Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/elementsoftheoryDOreidrich The Elements of the Theory of Algebraic Numbers BY LEGH WILBER REID Professor of Mathematics in Haverford College WITH AN INTRODUCTION BY DAVID HILBERT Professor of Mathematics in the University of Gottingen OF "HE UNIVERSITY OF Neto ¥orft THE MACMILLAN COMPANY 1910 «£*£%, Copyright 1910 By Legh Wilber Reid PRESS OF the new Era printing compan* lancaster. pa TO MY WIFE. PREFACE. It has been my endeavor in this book to lead by easy stages a reader, entirely unacquainted with the subject, to an appreciation of some of the fundamental conceptions in the general theory of algebraic numbers. With this object in view, I have treated the theory of rational integers more in the manner of the general theory than is usual, and have emphasized those properties of these integers which find their analogues in the general theory. The same may be said of the general quadratic realm, which has been treated rather as an example of the general realm of the nth degree than simply as of the second degree, as little use as was possible, without too great sacrifice of simplicity, being made of the special properties of the quadratic realm in the proofs. The theorems and their proofs have therefore been so formulated as to be readily extendable, in most cases, to the general realm of the 11th. degree, and it is hoped that a student, who wishes to continue the study of the subject, will find the reading of works on the general theory, such as Hilbert's Bericht iiber die Theorie der Algebraischen Zahlkorper, rendered easier thereby. The realm k ( V — I ) has been discussed at some length with two objects in view ; first, to show how exactly the theorems relating to rational integers can be carried over to the integers of a higher realm when once the unique factorization theorem has been estab- lished; and second, to illustrate, by a brief account ofr-Gauss' work in biquadratic residues, the advantage gained by widening our field of operation. The proofs of the theorems relating to biquadratic residues have necessarily been omitted but the examples given will make the reader acquainted with their content. The realms &(V — 3) and &(V 2 ) have been briefly discussed in order to introduce the reader to modifications which must be made in our conceptions of integers and units. In &(y — 5), the failure of the unique factorization law is shown and its restoration in terms of ideal factors is foreshadowed. VI PREFACE. References have been given more with a view to aiding the student in continuing his study of the subject than to pointing out the original source of a theorem or concept. The author has adopted the term " realm " as the equivalent of korper, corpus, campus, body, domain and field, as it has the advantage, he believes, of not having been used in any other branch of mathematics. It is suggested by Gauss' use of the term " Biirgerrecht " in connection with his introduction of the integers of k(y/ — i) as his field of operation (see p. 218). Many numerical examples have been given, especially in cases involving ideals, and it is hoped that through them the student may attain some familiarity with the methods of reckoning with algebraic numbers. The fact that the earlier discoveries in the theory of numbers were made inductively inspires the belief that such discoveries may also be made in the higher theory, if a sufficient amount of numerical material be at hand. The following is a list of the principal authorities that have been consulted, the abbreviations used in citation being given. The lectures of Professor Hilbert, mentioned above, the use of which he kindly allowed me. Bachmann: Die Lehre von der Kreis- theilung; Elemente der Zahlentheorie ; Niedere Zahlentheorie ; Allegmeine Arithmetik der Zahlenkorper. Borel et Drach: Le- cons sur la Theorie des Nombres et Algebra. Cahen: Elements de la Theorie des Nombres, cited as Cahen. Cayley: Encyclo- paedia Britannica, 9th ed., Vol. XVII, pp. 614-624. Chrystal: Algebra. Dirichlet-Dedekind : Vorlesungen iiber Zahlentheorie, 4th ed, cited as Dirichlet-Dedekind. Gauss : Disquisitiones Arith- meticae, Works, Vol. I; Theoria Residuorum Biquadraticorum, Commentatio Prima, Commentatio Secunda, Works, Vol. II. Hilbert: Bericht iiber die Theorie der Algebraischen Zahlkorper, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. IV, cited as Hilbert: Bericht. Kronecker: Vorlesungen iiber Zahlen- theorie. Laurent: Theorie des Nombres, Ordinaires et Alge- briques. Mathews : Theory of Numbers, cited as Mathews ; also Encyclopaedia Britannica, Supplement, Vol. XXXI. Minkowski : Geometrie der Zahlen ; Diophantische Approximationen. H. J. PREFACE. Vll S. Smith : Report on the Theory of Numbers, Collected Mathe- matical Papers, Vol. I, pp. 38-364, cited as H. J. S. Smith. Tsche- byscheff : Theorie der Congruenzen. Weber : Algebra. Wertheim ; Elemente der Zahlentheorie ; Anfangsgriinde der Zahlenlehre. In conclusion, I wish to express my most sincere thanks to Professor Hilbert for having given me my first interest in the subject of the theory of numbers by his lectures, which I attended in the winter semester, 1897-98, at Gottingen, for his continued interest in my work, and for his great kindness in writing an introduction to this book. I desire also to acknowledge my indebtedness to Professor James Harkness of McGill Uni- versity for many helpful suggestions, and to the late Professor J. Edmund Wright of Bryn Mawr College and my colleague Professor W. H. Jackson for valuable assistance with the proof sheets. Legh W. Reid. Haverford College. CONTENTS. INTRODUCTION. CHAPTER I. Preliminary Definitions and Theorems. § I. Algebraic numbers. Algebraic integers. Degree of an algebraic number I § 2. Algebraic number realms 3 §3. Generation of a realm 3 § 4. Degree of a realm. Conjugate realm. Conjugate numbers 5 § 5. Forecast of remaining chapters 5 CHAPTER II. The Rational Realm. Divisibility of Integers. § 1. Numbers of the rational realm 7 § 2. Integers of the rational realm 7 § 3. Definition of divisibility 8 § 4. Units of the rational realm 8 § 5. Rational prime numbers 9 § 6. The rational primes are infinite in number 10 § 7. Unique factorization theorem 12 - § 8. Divisors of an integer 23 § 9. Determination of the highest power of a prime, p, by which m ! is divisible 26 * tn ' § 10. The quotient , — — 1 where m, = a + &+••• + k, is an a ! ! • • • k ! integer 28 CHAPTER III. The Rational Realm. Congruences. § 1. Definition. Elementary theorems 31 § 2. The function ( m) 2>7 § 3. The product theorem for the <P- function 45 § 4. The summation theorem for the 0-f unction 46 y ix X CONTENTS. § 5. Discussion of certain functional equations and another derivation of the general expression of <t>{m) in terms of in 49 § 6. 0-f unctions of higher order 54 §7. Residue systems formed by multiplying the numbers of a given system by an integer prime to the modulus 56 § 8. Fermat's Theorem as generalized by Euler 57 §9. Congruences of condition. Preliminary discussion 59 § 10. Equivalent congruences 62 §11. Systems of congruences. Equivalent systems 64 § 12. Congruences in one unknown. Comparison with equations .... 66 § 13. Congruences of the first degree in one unknown 68 § 14. Determination of an integer that has certain residues with respect to a given series of moduli 70 § 15. Divisibility of one polynomial by another with respect to a prime modulus. Common divisors. Common multiples 76 § 16. Unit and associated polynomials with respect to a prime modulus. Primary polynomials JJ § 17. Prime polynomials with respect to a prime modulus. Deter- mination of the prime polynomials, mod p, of any given degree. 78 § 18. Division of one polynomial by another with respect to a prime modulus ,. 79 _i§ 19. Congruence of two polynomials with respect to a double modulus. 81 § 20. Unique fractionization theorem for polynomials with respect to a prime modulus 82 §21. Resolution of a polynomial into its prime factors with respect to a prime modulus 87 § 22. The general congruence of the nth degree in one unknown and with prime modulus 88 § 23. The congruence x4* m ) — 1^0, mod m 90 § 24. Wilson's Theorem , 91 § 25. Common roots of two congruences 92 §26. Determination of the multiple roots of a congruence with prime modulus 93 § 27. Congruences in one unknown and with composite modulus .... 95 § 28. Residues of powers 98 § 29. Primitive roots 104 § 30. Indices 105 § 31. Solution of congruences by means of indices 108 § 32. Binomial congruences no §33. Determination of a primitive root of a given prime number 112 § 34. The congruence x n === b, mod p. Euler's criterion 114 CONTENTS. XI CHAPTER IV. The Rational Realm. Quadratic Residues. § i. The general congruence of the second degree with one unknown. 119 § 2. Quadratic residues and non-residues 121 § 3. Determination of the quadratic residues and non-residues of a given odd prime modulus 124 § 4. Legendre's Symbol 127 § 5. Determination of the odd prime moduli of which a given integer is a quadratic residue 128 § 6. Prime moduli of which — 1 is a quadratic residue 128 §7. Determination of a root of the congruence x 2 ^ — 1, mod p, (p = 4m -f- 1) by means of Wilson's Theorem 129 § 8. Gauss' Lemma 130 § 9. Prime moduli of which 2 is a quadratic residue 133 § 10. Law of reciprocity for quadratic residues 135 §11. Determination of the value of (a/p) by means of the quadratic reciprocity law, a being any given integer and p a prime 144 § 12. Determination of the odd prime moduli of which a given positive odd prime is a quadratic residue 145 § 13. Determination of the odd prime moduli of which any given • integer is a quadratic residue 147 § 14. Other applications of the quadratic reciprocity law 149 CHAPTER V. The Realm k(i). § 1. Numbers of k(i). Conjugate and norm of a number 155 § 2. Integers of k (»") 157 § 3. Basis of k (i) 159 § 4. Discriminant of k (i) 161 § 5. Divisibility of integers of k (i) 162 § 6. Units of k (i) . Associated integers 163 § 7. Prime numbers of k (*) 165 § 8. Unique factorization theorem for k(i) 167 § 9. Classification of the prime numbers of k(i) 177 §10. Factorization of a rational prime in k(i) determined by the value of (d/p) 179 § 11. Congruences in k(i) 180 § 12. The 0-f unction in k (t) 185 § 13. Residue systems formed by multiplying the numbers of a given system by an integer prime to the modulus 188 §14. The analogue for k(i) of Fermat's Theorem 189 § 15. Congruences of condition 190 Xll CONTENTS. § 16. Two problems 191 § 17. Primary integers of k(i) 193 §18. Quadratic residues and the quadratic reciprocity law in k(i).. 196 § 19. Biquadratic residues 205 CHAPTER VI. The Realm fc(V-— 3)- § 1. Numbers of fc( V — 3 ) 218 § 2. Integers of k (V — 3) 219 §3- Basis of k(\/ — 3) ^^ 220 § 4. Conjugate and norm of an integer of & (V — 3) 221 § 5. Discriminant of £ (V — 3) 221 § 6. Divisibility of integers of &(V — 3) 221 §7. Units of &(V — 3). Associated integers 222 §8. Prime numbers of &(V — 3) 223 § 9. Unique factorization theorem for k (V —3) 226 § 10. Classification of the prime numbers of £(V — 3) 227 §11. Factorization of a rational prime in fc(V — 3) determined by the value of (d/p) 229 §12. Cubic residues 230 CHAPTER VII. The Realm k(\f2). § 1. Numbers of &(V 2 ) 231 §2. Integers of &(V 2 ) 231 § 3. Discriminant of k{ V 2 ) 232 § 4. Divisibility of integers of £(V" 2 ) 232 § 5. Units of &(V 2 ). Associated integers 232 § 6. Prime numbers of fc(V 2 ) 235 §7. Unique factorization theorem for &(V 2 ) 236 §8. Classification of the prime numbers of k(\j2) 238 §9. Factorization of a rational prime in &(V 2 ) determined by the value of (d/p) 240 § 10. Congruences in &(V 2 ) 240 §11. The Diophantine equations x 2 — 2y 2 = ± 1, x 2 — 2y 2 =±/>, and x 2 — 2y 2 = ± m 240 CHAPTER VIII. The Realm £(\A— 5)- § 1. Numbers of fc(V — 5) 245 § 2. Integers of fc(V — 5 ) 245 §3. Discriminant of fc(V — 5) 245 CONTENTS. Xlll § 4. Divisibility of integers of k( V — 5) 245 § 5. Units of k ( \A— 5) • Associated integers 246 §6. Prime numbers of fc(V — 5) 246 §7. Failure of the unique factorization theorem in &(V — 5). Intro- duction of the ideal 247 § 8. Definition of an ideal of k(y/ — 5) 257 § 9. Equality of ideals 258 § 10. Principal and non-principal ideals 260 § 1 1. Multiplication of ideals 261 § 12. Divisibility of ideals 263 § 13. The unit ideal 263 § 14. Prime ideals 263 § 15. Restoration of the unique factorization law in terms of ideal factors 265 CHAPTER IX. General Theorems Concerning Algebraic Numbers. § 1. Polynomials in a single variable , . 268 § 2. Numbers of a realm 271 CHAPTER X. The General Quadratic Realm. § 1. Number defining the realm 280 § 2. Numbers of the realm. Conjugate and norm of a number. Primitive and imprimitive numbers 281 § 3. Discriminant of a number 284 § 4. Basis of a quadratic realm 284 § 5. Discriminant of the realm 287 § 6. Determination of a basis of k(yjm) 289 CHAPTER XI. Ideals of a Quadratc Realm. § 1. Definition. Numbers of an ideal 293 §2. Basis of an ideal. Canonical basis. Principal and non-principal ideals 294 § 3. Conjugate of an ideal 301 § 4. Equality of ideals 302 § 5. Multiplication of ideals 302 § 6. Divisibility of ideals. The unit ideal. Prime ideals 303 § 7. Unique factorization theorem for ideals 305 XIV CONTENTS. CHAPTER XII. Congruences whose Moduli are Ideals. § i. Definition. Elementary theorems 323 § 2. The norm of an ideal. Classification of the numbers of an ideal with respect to another ideal 326 § 3. Determination and classification of the prime ideals of a quadratic realm 339 § 4. Resolution of any given ideal into its prime factors 348 § 5. Determination of the norm of any given ideal 351 § 6. Determination of a basis of any given ideal 351 §7. Determination of a number a of any ideal a such that (a) /a is prime to a given ideal, m 356 § 8. The 0-f unction for ideals 358 § 9. Residue systems formed by multiplying the numbers of a given system by an integer prime to the modulus 367 §10. The analogue for ideals of Fermat's Theorem 368 § 1 1. Congruences of condition 369 § 12. Equivalent congruences 372 § 13. Congruences in one unknown with ideal moduli 374 § 14. The general congruence of first degree with one unknown .... 375 § 15. Divisibility of one polynomial by another with respect to a prime ideal modulus. Common divisors. Common multiples 380 § 16. Unit and associated polynomials with respect to a prime ideal modulus. Primary polynomials 380 § 17. Prime polynomials with respect to a prime ideal modulus. De- termination of the prime polynomials, mod p, of any given degree 381 § 18. Division of one polynomial by another with respect to a prime ideal modulus 382 § 19. Unique f ractorization theorem for polynomials with respect to a prime ideal modulus 382 §20. The general congruence of the nth. degree in one unknown and with prime ideal modulus 385 §21. The congruence #*( m > — i==o, mod m 387 §22. The analogue for ideals of Wilson's Theorem 388 § 23. Common roots of two congruences 389 § 24. Determination of the multiple roots of a congruence with prime ideal modulus 390 § 25. Solution of congruences in one unknown and with composite modulus 391 § 26. Residues of powers for ideal moduli 392 §27. Primitive numbers with respect to a prime ideal modulus 398 § 28. Indices 398 § 29. Solution of congruences by means of indices 400 CONTENTS. XV CHAPTER XIII. The Units of the General Quadratic Realm. § i. Definition 403 § 2. Units of an imaginary quadratic realm 404 § 3. Units of a real quadratic realm 405 § 4. Determination of the fundamental unit 420 § 5. Pell's Equation 423 CHAPTER XIV. The Ideal Classes of a Quadratic Realm. § 1. Equivalence of ideals 427 § 2. Ideal classes 432 § 3. The class number of a quadratic realm 434 Index 452 ERRATA. Page 2.J, line 5, for "f x n read "p." Page 172, line 5, for 'V 3 = — 4 — p" read " fi s = 2. " p o Page 344, line 10, for P o " o I read p-\ 11 2 Page 356, line 29, for "(7, 3 + V~ 5) " read "(7, 3-V-5)." Page 392, line 3 of fine print, for "of (x) " read "f(x)." INTRODUCTION. • Die Zahlentheorie ist ein herrlicher Bau, erschaffen und auf- gefiihrt von Mannern die zu den glanzendsten Forschern im Bereiche der mathematischen Wissenschaften gehoren: Fermat, Euler, Lagrange, Legendre, Gauss, Jacobi, Dirichlet, Hermite, Kummer, Dedekind und Kronecker ; Alle diese Manner haben in den begeistersten Worten ihrer hohen Meinung uber die Zahlen- theorie Ausdruck gegeben und bis heute giebt es wohl keins Wissenschaft, von deren Ruhme ihre Jiinger so erfiillt sind, wie von der Zahlentheorie. Man preist an der Zahlentheorie die Einfachheit ihrer Grundlagen, die Genauigkeit ihrer Begriffe und die Reinheit ihrer Wahrheiten; man ruhmt sie als das Vorbild fur die anderen Wissenschaften, als die tiefste unversiegbare Quelle aller mathematischen Erkenntniss und als reiche Spenderin von Anregungen fur andere mathematische Forschungsgebietc wie Algebra, Funktionentheorie, Analysis und Geometric Dazu kommt, dass die Zahlentheorie vom Wechsel der Mode unab- hangig ist und dort nicht wie oft in anderen Wissensgebieten, bald die eine Auffassung oder Methode iibermassig sich auf- baus§ht, bald zu anderer Zeit unverdiente Zuriicksetzung erf ahrt ; in der Zahlentheorie ist oft das alteste Problem noch heute modern, wie ein echtes Kunstwerk aus der Vergangenheit. Und dennoch ist jetz wie friiher wahr, wo ruber Gauss und Dirichlet klagten, dass nur eine geringe Anzahl von Mathe- matikern zu einer eingehenden Beschaftigung mit der Zahlen- theorie und zu einem vollen und freien Genusse ihrer Schonheit gelangt. Zumal ausserhalb Deutschlands und unter der heran wachsenden mathematischen Jugend ist arithmetisches Wissen nur wenig verbreitet. Jeder Liebhaber der Zahlentheorie wird wunschen, dass die Zahlentheorie gleichmassig ein Besitz aller Nationen sei und gerade besonders unter der jungen Generation, der die Zukunft xvii XV111 INTRODUCTION. gehort, Pflege und Verbreitung finde. Das vorliegende Buch steckt sich dieses Ziel : Moge es dasselbe erreichen, indem es nicht nur dazu beitrage, dass die Elemente der Zahlentheorie Gemein- gut aller Mathematiker werden, sondern, indem es auch zugleich als Einfuhrung und Erleichterung zum Studium der darin ge- nannten Originalwerke diene, sowie zur selbstandigen Betha- tigung der Zahlentheorie anrege. Bei der liebevollen Vertiefung des Verfassers in die Zahlentheorie und bei dem hingebenden Verstandniss, mit dem der Verfasser in das Wesen derselben eingedrungen ist, durfen wir auf die Erfullung dieses Wunsches bauen. David Hilbert. Gottingen, io, Marz, 1907. TRANSLATION. The theory of numbers is a magnificent structure, created and developed by men who belong among the most brilliant investigators in the domain of the mathematical sciences : Fermat, Euler, Lagrange, Legendre, Gauss, Jacobi, Dirichlet, Hermite, Kummer, Dedekind and Kronecker. All these men have expressed their high opinion respecting the theory of numbers in the most enthusiastic words and up to the present there is indeed no science so highly praised by its devotees as is the theory of numbers. In the theory of numbers, we value the simplicity of its foundations, the exactness of its conceptions and the purity of its truths ; we extol it as the pattern for the other sciences, as the deepest, the inexhaustible source of all mathematical knowledge, prodigal of incitements to investigation in other departments of mathematics, such as algebra, the theory of func- tions, analysis and geometry. Moreover, the theory of numbers is independent of the change of fashion and in it we do not see, as is often the case in other depart- ments of knowledge, a conception or method at one time given undue prominence, at another suffering undeserved neglect; in the theory of numbers the oldest problem is often to-day modern, like a genuine work of art from the past. Nevertheless it is true now as formerly, a fact which Gauss and Dirichlet lamented, that only a small number of mathematicians busy themselves deeply with the theory of numbers and attain to a full enjoyment of its beauty. Especially outside of Germany and among the younger mathematicians arithmetical knowledge is little disseminated. Every devotee of the theory of numbers will desire that it shall be equally a possession of all nations and be cultivated and spread abroad, especially among the younger generation to whom the future INTRODUCTION. XIX belongs. Such is the aim of this book. May it reach this goal, not only by helping to make the elements of the theory of numbers the common property of all mathematicians, but also by serving as an introduction to the original works to which reference is made, and by inciting to inde- pendent activity in the field of the theory of numbers. On account of the devoted absorption of the author in the theory of numbers and the comprehensive understanding with which he has penetrated into its nature, we may rely upon the fulfilment of this wish. CHAPTER I. Preliminary Definitions and Theorems. § i. Algebraic Numbers. Algebraic Integers. Degree of an Algebraic Number. It will be assumed in this book that the complex number system has been built up and that the laws to which the four fundamental operations of algebra are subject have been demonstrated to hold when these operations are performed upon any numbers of this system. We shall occupy ourselves with certain properties of a special class of these numbers, known as algebraic numbers, these prop- erties flowing in the greater part from the relation in which two numbers stand to one another when one is said to be a divisor of the other. We proceed to define an algebraic number. A number, a, is said to be an algebraic number when it satisfies an equation of the form x n -f a 1 x n ~ x + • • • + On_ t x + a n = o i) where a lt a 2 , • • •, On are rational numbers. We shall call an equa- tion of form i) a rational equation. The simplest algebraic numbers are evidently the rational numbers. An algebraic num- ber is said to be an algebraic integer or briefly an integer, when it satisfies an equation of the form i) whose coefficients, a lt a 2 , • • •, a n , are rational integers. The simplest algebraic integers are the positive and negative natural numbers. An algebraic number, a, evidently satisfies an infinite number of rational equations, for if a satisfy i), it also satisfies any equation formed by multiplying i) by an integral function of x of the form #• + b x x m ~ x + • • • + b m _ x x + b m , where b lt - ■ •, b m are rational numbers, and this equation will be of the form i). There will be however among all these rational 2 PRELIMINARY DEFINITIONS AND THEOREMS. equations satisfied by a, one and only one of lowest degree, /. For suppose that a satisfied two different rational equations of the /th degree, / being the degree of the rational equation of lowest degree satisfied by a, and let these equations be x l -f a^x 1 - 1 + • • • + ai = o 2) x i + biX i-i + --. + bi = o 3) Then a will satisfy the equation formed by subtracting 3) from 2) ; that is, (a x — b^x 1 - 1 -f- •• • + a\ — bi = o > 4) Unless 4) be identically zero, a satisfies a rational equation of degree lower than the /th, which is contrary to our original sup- position. Therefore 4) is identically o, and 2) and 3) are the same equation. Hence a satisfies only one rational equation of the /th degree. This equation is irreducible ; that is, its first member can not be resolved into factors of lower degree in x, with rational coeffi- cients ; for if x 1 + aiX 1-1 + . . . + a , = ( x n + blX n-i + . . . + bh ) X (^ + <vtr*- 1 + --- + c k ), where b lt •••, b h , c x , •••, c fc are rational numbers, a would satisfy one of the rational equations x h + b^- 1 -f- 1- b h = o ; x k + c^v*- 1 + • • • + c k = o. This is, however, impossible since both of these equations are of lower degree than the /th. Hence the rational equation of lowest degree, which a satisfies, is irreducible. If / be the degree of this equation, a is said to be an algebraic number of the /th degree. Theorem i. If a be an algebraic number , f x ( x ) =x l + a x x l - x + ••• + fli==d the single rational equation of lowest degree which a satisfies, and f 2 (x) = x m + b^v™- 1 + • • • + b m = o any other rational, equation satisfied by a, then f x (x) is a divisor off 2 (x). PRELIMINARY DEFINITIONS AND THEOREMS. 3 We can always put f 2 (x) in the form where f^ix) and f 4 (x) are rational integral functions of x whose coefficients are rational integers and f 4 (x) of lower degree than f 1 (x). Substituting a for x in 2) we have / 2 (a)=/ 3 (a)-/ 1 (a)+/4(a), whence, since f 2 (a)=o, and /i(a)=o, f*(.CL)=o\ that is, unless / 4 (^r) is identically o, a will satisfy a rational equation, / 4 (^) =0 of lower degree than the /th. But this is contrary to our original hypothesis. Hence f 4 (x) is identically zero, and f x (x) is there- fore a divisor of f 2 (x). />,! J- We shall see later (Chap. II, Th. 4) that the rational equation of lowest degree which an algebraic number, a, satisfies, deter- mines the question whether or not a is an algebraic integer; that is, that the coefficients of the single rational equation of lowest degree, which an algebraic number, a, satisfies, shall be integers, is a necessary as well as sufficient condition for a to be an alge- braic integer. § 2. Algebraic Number Realms. A system of algebraic numbers is called a number realm or briefly a realm, if the sum, difference, product and quotient of every two numbers of the system, excluding division by o, are numbers of the system; that is, if the system is invariant with respect to these four operations. The simplest example of a realm is the system of all rational numbers, for evidently the sum, difference, product and quotient of any two rational numbers are rational numbers. Another ex- ample is the system of numbers of the form x-\- y y — T > where x and y take all rational values. For the sum, difference, product and quotient of any two of these numbers are numbers of this form. § 3. Generation of a Realm. If a be any algebraic number, the system consisting of all num- bers, which can be formed by repeated performance upon a of the 4 PRELIMINARY DEFINITIONS AND THEOREMS. four fundamental reckoning operations, that is, the system con- sisting of all rational functions of a with rational coefficients, will be a realm. For the sum, difference, product and quotient of any two ra- tional functions of a are rational functions of a and hence num- bers of the system. We say that a generates this realm. We say also that a defines the realm and denote the latter by k(a). The rational realm can be generated by any rational number, a ; for a divided by a gives I, and from I by repeated additions and subtractions of I, we can obtain all rational integers, and from them by division all rational fractions. As the number defining the rational realm we generally take i, thus denoting the realm by k(i). More usually, how- ever, the rational realm is denoted by the letter R. The realm given as the second example in the last paragraph can be generated by V — J ; f° r V — ! divided by V — T gives i, and from I we can generate the rational realm and then by multiplying V — I by all rational numbers in turn and adding to each of these products each rational number in turn, we obtain all numbers of the form x -f- yy/ — i, where x and y take all rational values. This realm is therefore denoted by k( V — i). We have seen in the last example that among the numbers of k ( V — i ) are found all the numbers of the rational realm. It may be easily seen that this is true of every realm ; that is, every realm contains R ; for if o> be any number, w divided by w gives i, and from i we can generate R. It is well to observe that, although V — i is the number which most conveniently defines k ( V — I ) and is indeed the one usually selected, it is not the only number that will serve this purpose. We see, on the contrary, that this realm can be generated by any number of the form a -f- b V — I where a and b are rational numbers, and b =f= o ; that is, k ( V — 1 ) and k{a-\-by/ — 1) are identical; for since k(a-\-by/ — 1) con- tains R, it contains a and b and hence -^ — ^-r , = y/ — 1. Therefore k(a-\-by/ — 1) contains all numbers of k(y/— Y). Moreover since &(V — 1) contains a-\-b\/ — 1, it contains all PRELIMINARY DEFINITIONS AND THEOREMS. 5 numbers of k{a-\-b\/ — i). Hence k ( V — I ) is identical with k(a-\-by — 1). It may be shown similarly that any realm may be defined by any one of an infinite number of its num- bers; as, for example, if a be any algebraic number, k(a) and k(a-\-ba), where a and b are rational numbers, and &=f=o are identical. A realm may be generated by any number of algebraic numbers. If a, /?, • • •, X are a finite number of algebraic numbers, the system consisting of all rational functions of these numbers with rational coefficients is a realm which we denote by k(a, fi, ■••, A). It can be shown, however, that in every realm k(a, /?, ••••, A) we can find a number 6 such that k(a, /?, •••, \)=k{6). We shall not prove this, as all realms discussed in this book will be defined by a single number. § 4. Degree of a Realm. Conjugate Realms. Conjugate Numbers. If the rational equation of lowest degree which a satisfies be x n + a^x n - x -f • • • + a n = o 1) then k(a) is said to be of the nth. degree. That is, the degree of a_realm is the degree of the number defining the realm. Thus &(V — 1) is of the second degree, since the rational equation of lowest degree which V — 1 satisfies is x 2 + 1 = o. Likewise £(1/2) is of the third degree. There is evidently only one realm of the first degree k(i), but an infinite number of all other de- grees. If the remaining roots of 1 ) be a', a", • • •, a (n_1) , then n — 1 realms k(a'), k(a") f •••, ^(a (n_1) ) are called the conjugates of If 6 be any number of k(a), it is a rational function of a, which we may denote by r{a). Then 0' = r(a')> 0" = r(a"), •••, (n_1) =r(a (n ~ 1) ), which are derived from by the substitutions a: a', a: a", •••,a:a (n_1) , are called the conjugates of 0. § 5. Forecast of Remaining Chapters. We shall consider now several special realms. In each we shall find an infinite number of algebraic integers, the study of whose properties in their mutual relations will be our task. It will be 6 PRELIMINARY DEFINITIONS AND THEOREMS. observed that the properties of an integer depend upon the realm in which it is considered to lie. Thus the integer 5 is unfavor- able in R and in k( V — 3), but in k(y/ — 1) it is the product of two integers, 2 -f- V — 1 and 2 — V — 1. The realms will be taken up in the order of their degrees. That is, the first to be studied will be R, which is, as has been already said, the only realm of the first degree. We shall then take up in turn four special examples of quadratic realms, £(V — 1), &(V — 3), £(V 2 ) and &(V — 5). In the cases of &(V — 1), fc(V — 3) and k(\/2), we shall see that, with the introduction of a few new conceptions, the integers of these realms obey in their relations to each other laws almost identical with those governing the integers of R. In the case of &(V — 5) we shall observe an important differ- ence, and at first sight it will seem that our old laws have no analogues in this realm. By the introduction, however, of the conception of the ideal number not only will the difficulties of this particular realm be overcome, but we shall be able to establish in terms of these jdeal numbers general laws for the mutual rela- tions of the integers of any quadratic realm, which are analogous to those already found for the integers of the special realms ex- amined. Furthermore the larger part of the theorems proved for the integers of the general quadratic realm hold for the in- tegers of a realm of any degree whatever. CHAPTER II. The Rational Realm. divisibility of integers. § i. The Numbers of the Rational Realm. The rational realm consists of the system of rational numbers, any one of which, except o, may be taken to define it. It is usually denoted by k(i) or simply R. The absolute value of a number, m, of R is m taken positively and is denoted by | m \ . Thus l±5l=5- The absolute value of a number is used when the result of an enumeration is to be expressed as a function of this number. § 2. Integers of the Rational Realm. The positive and the negative rational integers are evidently integers of R, for they satisfy equations of the form ;r-|-a = o, where a is a rational integer. The sum, difference and product of any two rational integers are seen to be integers. The ques- tion will at once be asked, are these all the numbers of the rational realm which are algebraic integers under the definition given of an algebraic integer (Chap. I, § i). That is, although a rational fraction, b/c, where b is not divisible by c evidently cannot satisfy an equation of the form x-\-a = o, where a is a rational integer, we have not yet shown that b/c cannot satisfy an equation of higher degree than the first and of the form x n + a x x n ~ x -f • • • -+- a n = o, where a x ,a 2 , ••-,(!„ are rational integers. To show this, it is necessary to prove first that a rational integer can be resolved in one and only one way into prime factors. Therefore, until we have proved this theorem, the integers with which we are dealing should be looked upon as merely the ordi- nary rational integers. When we have proved the above theorem 7 8 THE RATIONAL REALM INTEGERS. we shall see that the system of rational integers and the system of integers of R are coextensive. § 3. Definition of Divisibility. An integer, a, is said to be divisible by an integer, b, when there exists an integer, c, such that a=bc; then b and c are said to be divisors, or factors, of a and a is said to be a multiple of b and c. Furthermore, a is said to be resolved into the factors b and c, or to be factored. We have, as direct consequences of the definition of divisibility and the fact that the sum, difference and product of any two integers are integers, the following: i. If a be a multiple of b, and b a multiple of c, a is a multiple of c. For since a is a multiple of b, we have a = a 1 b, and since b is a multiple of c, b = b x c. From which it follows that a = a x b x c. Hence a is a multiple of c. In general if each integer of a series a, b, c, d, • • •, be a multiple of the one next following, each integer is a multiple of all that follow it ; that is, if a be a multiple of b, b a multiple of c, c a multiple of d, etc., a is a mul- tiple of b, c, d, • - •, b a multiple of c, d, • • -, etc. ii. // two integers a and b be multiples of an integer c, a -f- b and a — b are multiples of c. If two or more integers a, b, c, ••• be each divisible by an integer m, m is said to be a common divisor or common factor of a, b, c, •• •. If an integer, m, be a multiple of two or more integers, a, b, c, •••, m is said to be a common multiple of a, b, c,- -, 1 § 4. Units of the Rational Realm. There are two integers, 1 and — 1, which are divisors of every rational integer and they are the only rational integers that enjoy this property. We call 1 and — 1 the units of R. Any integer which is divisible by m is also divisible by — m; hence any two integers which differ only by a unit factor are considered as identical in all questions of divisibility. We say throughout this book the letters of the Latin alphabet will always denote rational numbers, unless there be a direct statement to the contrary. THE RATIONAL REALM INTEGERS. 9 that two such integers are associated, and call either one the asso- ciate of the other. Two integers, a and b, each of which divides the other, are associates, for if a = cb and b = da where c and d are integers, then cd=i, and hence c=±i. Two integers whose absolute values are the same are evidently associates. For the sake of generality we consider an integer as associated with itself. Thus the associates of 5 are 5 and — 5 since 5 = 1-5 and— 5— —1-5. The factorizations of 30, 30 = 2-3-5, = — 2- — 3-5, = — 2-3* — 5, = 2 — 3- — 5, are looked upon as identical, since they differ only by the replace- ment of one or more of the factors by their associates. Two integers with no common divisors other than units are said to be prime to each other. Under this definition the units are considered prime to every integer including themselves. if i'«i-i»i a and b are associates, and it follows therefore that if a be prime to b | a | =(= | b | unless a and b be units. A system of integers such that no two of them have common divisors other than the units are said to be prime each to each. § 5. Rational Prime Numbers. Any integer, p, that is not a unit and that has no divisors other than p and — p, 1 and — 1, that is, than its associates and the units, is called a prime number or, briefly, a prime. The units are not considered to be prime numbers, because many of the theorems relating to prime numbers will be found not to hold for them. IO THE RATIONAL REALM INTEGERS. Every integer, m, with divisors other than m, — m, I, — I is called a composite number. We can obtain the positive prime numbers less than any given positive integer, m, as follows : The only even one is 2. We write down then the odd integers smaller than m, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, •••, and remove from the series those which are composite. To do this strike out, counting from 3, the 3d, 6th, 9th, • • • integers ; that is, 9, 15, 21, ••*. Then counting from 5, strike out the 5th, 10th, ••• integers; that is, 15, 25, •••, counting integers already struck out, and in general, if p be the smallest integer not struck out, excluding those whose multiples have been struck out, we strike out the pth, 2pth, 3pth, • • • integers, counting from p ; that is, all multiples of p except p. The integers not struck out are the positive primes smaller than m. This method is known as the Sieve of Eratosthenes. It is, however, not necessary to carry out the process for every prime, p, smaller than m ; for every composite number, m lf smaller than p 2 , will have been struck out as a multiple of a prime smaller than p, since if m 1 be less than p 2 , it contains as a factor a prime less than p. The greatest value of p for which the process must be car- ried out is therefore the greatest prime not greater than s/m. The positive primes less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Ex. 1. Show that every rational prime, except 2, is either of the form 4« — 1 or 4*1 -\- 1. Ex. 2. Show that every rational prime, except 2 and 3, is either of the form 6n — 1 or 6n + 1. § 6. The Rational Primes are Infinite in Number. The proof of this theorem as given by Euclid (Elements, Book IX, Prop. 20) is the following : Let us suppose that there are only a finite number of positive primes, p being the greatest. Multiply these primes together and add 1 to the product, forming the number It is evident that N is not divisible by any of the primes 2, 3, 5, •••, p. Hence N is either a prime itself, or contains as a factor THE RATIONAL REALM INTEGERS. I I a prime greater than p. In either case there exists a prime greater than p, which contradicts our original assumption. Hence the number of rational primes is infinite. This proof of Euclid's tells us far more than merely that the rational primes are infinite in number, for if 2, 3, • • •, p be the n smallest positive primes it gives a limit, p-\-i to 2-y-p -\- 1, within which a prime greater than p must lie. To bring out clearly what has been proved we may state the theorem as follows : // 2, 2, '", P be the n smallest positive primes, then there is a prime greater than p among tlve numbers p -\-i, •••,2-3 -• p -\-i and consequently the rational primes are infinite in number. For example, 2, 3, 5, 7, being all the positive rational primes not greater than 7, there is certainly one prime greater than 7 among the numbers 8,9, •••,3:3"5 , 7-+ I. After it became known that the rational primes are infinite in number, the attention of investigators was turned to the question whether, if from the positive integers a series be selected which form an arithmetical progression, as for example 1, 5, 9, 13, •••, or 3, 7, 11, 15, ■••, there are in every such series an infinite number of primes. Proofs showing that this is true of the two series given will be found in this book. It is not difficult to prove also that there are an infinite number of primes of each of the forms 6n — 1, 6n -(- 1, and Sn -f- 5. 1 These are, however, only special cases of the general theorem that in every unlimited arithmetical progression, whose general term is a -f- nd, the first term a and the common difference, d, being prime to each other, there occur infinitely many prime num- bers. This theorem was first proved by Dirichlet (see D. D., 4th Ed., Sup. VI), but he did not give an interval within which a new prime must lie, as in the case of Euclid's proof. This omission was supplied by Kronecker in 1885. (See above reference.) Among problems relating to prime numbers which still await solution is first of all that known as the problem of the frequency of the primes. It consists in the determination of the number of 1 Kronecker: Vorlesungen iiber Mathematik; Part II, Vol. I, p. 438. Cahen: Theorie des Nombres, p. 318. 12 THE RATIONAL REALM INTEGERS. positive primes less than any given positive number m, that is, in the determination of the law which governs the distribution of the primes among the entire series of positive integers. Kronecker mentions two interesting theorems which are be- lieved to be true, although no proofs have yet been obtained. I. Every positive even integer can be represented as the sum of two positive prime numbers (2 excepted). This theorem was first stated by Goldbach, then by Waring. Kronecker remarks 1 that after testing this theorem for the even integers from 2 to 2000, it is observed that the number of possible representations of 2n in this form increases as n increases, which heightens the probability of correctness ; for example, we have 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 55 10 = 3 + 7, 5 + 5; 12 = 5+7; 14 = 3+11, 7 + 7', 16 = 3+13, 5+n; 18 = 5 + 13, 7 +11; etc. II. Every positive even integer can be represented in infinitely many ways as the difference of two positive primes. If the truth of this theorem be assumed and it be applied to the integer 2, we obtain the theorem : However far we may go in the series of positive primes, we shall always find primes which differ only by 2, that is, which lie as close as possible together. Natur- ally the frequency of such pairs of primes decreases the farther out we go in the series of positive integers. Among the first one hundred integers there are eight such pairs : 3, 5; 5, 7; H* 13; 17, 19; 2 9, 3i; 4i, 43; 59, 61; 71, 73; and among the second hundred seven : 101, 103 ; 107, 109 ; 137, 139 ; 149, 151 ; *79, 181 ; 191, 193 ; 197, 199. If we go sufficiently far in the series of positive integers we can find as great a number of successive integers as we please, no one of which is a prime, for none of the integers n! + 2, n! + 3, -",nl-\-n is a prime, since nl-^-i, i^n, is divisible by *; for example, 5 ! + 2, 5 ! + 3, 5 ! + 4, 5 ! + 5 are all composite numbers. §7. Unique Factorization Theorem. According to the definition, every composite number can be 1 Vorlesungen uber Math., Part II, Vol. I, p. 68. THE RATIONAL REALM INTEGERS. 13 resolved into the product of two factors, neither of which is a unit. One or both of these factors may be composite, and hence in turn resolvable into two factors, neither of which is a unit, and we can continue this process until we reach factors which are primes. It is evident that when one or both of the factors are composite, the resolution is not unique; for example, 210=14-15 = io-2i =6-35 = 2- 105 = 3-70= 5-42 = 7.30. We shall show that, when the resolution is continued until the factors are primes, it will be unique, considering associated factors as the same (see § 4) , and that such a resolution is always possible ; for example, 7 7 7 7 5; 210=14-15 =2-7-3-5 = 10-21 =2-5-3-7 = 6-35 =2-3 = 2-105 = 2-3- = 3-7o =3-2 = 5.42 =5-2. = 7-30 =7-2 that is, 210 can be represented in only one way as a product of prime numbers. To prove this theorem, upon which the whole theory of the rational integers depends, that is, that every rational integer can be represented in one and only one way as a product of prime numbers, we require the two following theorems : Theorem A. // a be any integer and b any integer different from 0, there exists an integer m such that \a — mb\<\b\ Let f^zm + r, where 01 is the integer nearest to - and hence | r | ^ J ; then m is the required integer, for m <i, whence, multiplying by | b |, \a — mb I < I 6 |. This theorem is equivalent to saying that we can divide a by b rr -me 14 THE RATIONAL REALM INTEGERS. so as to obtain a remainder less in absolute value than b, the quo- tient being m. There are, except when a is divisible by b, evi- dently two integers which satisfy the requirements of the theorem, one selected as above and another differing from the first by I ; for example, if a =12 and b = — 5, then |l2-(-2)(-s)|<|-5| and |i 2 _(_3)(_ S )|<|_5|; and hence both — 2 and — 3 satisfy the requirements of the theorem, -j- 2 being the integer selected as in the proof. Theorem B. // a and b be any two integers prime to each other, there exist two integers, x and y, such that ax -{-by = 1. If either a or b be a unit, the existence of the integers x, y^is evident. We shall now show that, if neither a nor b be a unit, the determination of x and y can be made to depend upon the determination of a corresponding pair of integers x x , y x for a pair of integers a lf b x prime to each other and such that one of them is less in absolute value than both a and b. Assume |&|<|a|, which evidently does not limit the generality of the proof. By Th. A there exists an integer m such that |a — mb\<\b\. Then b and a — mb are a pair of integers, a x , b x , prime to each other, and a — mb is less in absolute value than both a and b. If now two integers x x , y x exist such that a x x x + b x y x =fi; that is, bx x + (a — mb)y x = 1, we have ciy x -\-b(x x — my x ) = i, and hence x=y ti y = x 1 — my x . The determination of x x , y x for a x , b x may, if neither a x nor b x be a unit, be made to depend similarly upon that of x 2 , y 2 for a pair of integers a 2f b 2 prime to each other and such that one of them is less in absolute value than both a x and b x . By a continua- tion of this process, we are able always to make the determination of x and y depend eventually upon that of x n , y n for a pair of integers a n , b„, one of which is a unit. THE RATIONAL REALM INTEGERS. I 5 Since the existence of x n and y n is evident, the existence of x and y is proved. Ex. Let a ■ = 14, b = 9 ; then oj = 9, b 1 = 5, and the determination of x and ;y, so that 14^ + 9y = 1 2) depends upon the determination of x lf y u so that gx s + 53>i = 1. 3) We can make the determination of x u y t depend upon the determination of x 2 , y 2 for the pair of integers a 2 = 5, b 2 — — 1, but it is sufficient here to notice that x x xs — 1, y x = 2 satisfy 3) and hence ,r = y 1 = 2, y =#i — m^ = — 1 — 1 -2 = — 3 satisfy 2) . The problem of finding the two integers x and y is most easily solved by the method of continued fractions, but the form of proof here used to show the existence of x and y has been adopted as being more easily applicable to realms of higher degree. The proof given satisfies completely, however, the requirement which Kronecker considered should be imposed upon every existence proof in the Theory of Numbers (see below) ; that is, it furnishes a method by which in a finite number of steps the desired integers x, y can be found from the given ones a, b. Hensel says in his preface to Kronecker's "Lectures on the Theory of Numbers," " Kronecker consciously imposed upon the definitions and proofs of the general arithmetic a demand whose rigorous observance essentially distinguishes his exposition of the theory of numbers and algebra from almost all others. " He considered that one can and must so formulate each definition in this domain that by a finite number of trials it can be determined whether or not it is applicable to any proposed quantity. " Likewise a proof of the existence of a quantity is to be looked upon as rigorous only when it contains at the same time a method, by which the quantity, whose existence is proved, can be actually found. Kronecker was very far from throwing entirely aside a definition or proof which did not satisfy these high requirements, but he considered that something was still wanting and he held its completion in this direction to be an important task, by which our knowledge would be extended in an es- sential point." " He considered, moreover, that a formulation rigorous in this sense was in general of simpler form than another which did not satisfy this demand and he has in many cases shown by his lectures that this is the case." Cor. // a and b be any two rational integers, there exists a common divisor d of a and b such that every common divisor 1 6 THE RATIONAL REALM INTEGERS. of a and b divides d, and we can find two integers x and y such that ax -\-by = d. Let a = a x c, b = b x c, where a x and b x and prime to each other. By Theorem B two integers x and y exist such that a x x + b x y = i. i) Multiplying I ) by c, we have a x cx + b x cy = c ; that is ax -\-by = c. Every common divisor of a and b evidently divides c. Hence c is the divisor, d, sought. We call d the greatest common divisor of a and b. It is evident that two such divisors which are not associates cannot exist; for if d x , d 2 be two such divisors, then since from the definition d x must divide d 2 and d 2 must divide d lt d x and d 2 are associates. Any number of integers, a lt a 2 , •••, a n , possess a common di- visor which is divisible by all common divisors of these integers ; for let d x be the greatest common divisor of a lf a 2 as defined above. Then two integers, x x and x 2 , exist such that a \ x x + a 2 x 2 — ^1- Let now d 2 be the greatest common divisor of d x and a 3 . It is evident that d 2 is a common divisor of a x , a 2 , a 3 , and that two integers, y x , y 2 , exist such that d 1 y 1 + a 3 y 2 = d 2 , or a x x x y x + a 2 x 2 y x +a z y 2 = d 2 ; that is, three integers, z x , z 2 , z z , exist such that a x z x + a 2 z 2 + a 3 z 3 = d 2 , from which identity it is evident that every common divisor of a x> a 2} a 3 , divides d 2 . Proceeding similarly with d 2 and a 4 , then with their greatest THE RATIONAL REALM INTEGERS. 1 7 common divisor d 3 and a 5 , etc., we see finally that there exist n integers u lf u 2 , •-,u n such that a i M i + °2 u 2 ~f~ ' * " ~f" fl nW n = d, where d is a common divisor of a x , a 2 , • • *, a n . From this identity it is evident that every common divisor of a 1} a 2 , ... a-n divides d. We call d therefore the greatest common divisor of the n integers a x , a 2y • • •, a n . The common divisors of a system of integers are evidently the divisors of the greatest common divisor of the system. To find the greatest common divisor of n integers a 1} a 2 , • • •, o«, we find the greatest common divisor d x of a x and a 2 ; then the greatest common divisor of d x and a z , which will evidently be the greatest common divisor of a lt a 2 , a 3 . Proceeding in this manner we arrive finally at an integer d which is the greatest common divisor of all of the integers. In particular, if a^, a 2 , •••, a„ have the greatest common divisor I, we have a 1 u 1 + a 2 u 2 + • * * + &nU n = i. This corollary is usually known as the greatest common divisor theorem and can be proved independently of Theorem B which follows easily from it. The independent proof of the corollary depends upon Theorem A and the following simple theorem whose truth is obvious. If a = mb + r, then every integer which divides both a and b divides both b and r, and vice versa; that is, the common divisors of a and b are identical with the common divisors of b and r. By virtue of these two theorems we are able to substitute for the problem of finding the integer which is divisible by all common divisors of a and b (|&| = |fl|) the corresponding problem for the two integers b and r, where a = mb-\-r, and | r \ < | b |.* From Theorem A, it is evident that we can form a chain of identities, a = mb -f- r, b = mjr -f n, r = m 2 r t + r 2 , 1 Euclid : Elements, Book VII, Prop. 2. 2 I 8 THE RATIONAL REALM INTEGERS. in which | r | > | r x \ > \r 2 \, etc., arriving after a finite number of such steps, since the integers less in absolute value than a given integer are finite in number, at a remainder r n +i which is o, and hence r n -i = m n+1 r n Now from the theorem above it is evident that the common divisors of a and b are identical with the common divisors of b and r, and hence with those of r and n, and finally with those of r ni and r n . But r n is a common divisor of r%^ and r n and evidently is divisible by every common divisor of r n - x and r«. Hence rn is the desired common divisor of a and b ; that is, it is divisible by all the common divisors of a and b. Moreover, we can by means of the method of continued fractions ex- press d,=rn, in the form ax + by = d. 1 The greatest common divisor of two or more integers is seen to be the common divisor of greatest absolute value, there being only one such common divisor since, if | a | = | b |, then a and. b are associates. It is also, as we have seen from the proof of the above corollary, the common divisor such that the quotients ob- tained by dividing each of the integers by this divisor have no common divisor other than ± I. The reason why neither of these properties has been chosen for the definition of the greatest common divisor of two or more integers will appear later (see p. 252). An objection to the former of the two, which is the one usually employed is, however, immediately evident in that the idea of inequality is introduced, whereas the question is purely one of divisibility. Theorem C. // the product of two integers, a and b, be divis- ible by a prime number, p, at least one of the integers is divisible by p. Let ab — cp, and assume a not divisible by p. Then a and p have no common divisor, and there exist two integers, x and y, such that ax -\- py = 1. 1 ) x Cahen: p. 60. Bachman : Niedere Zahlentheorie, p. 107. Chrystal : Vol. II, p. 445. THE RATIONAL REALM INTEGERS. 1 9 Multiplying I ) by b, we have bax + bpy = b, and therefore (ex + by)p = b, where ex -\- by is an integer. Hence b is divisible by p. Cor. 1. // the product of any number of integers be divisible by a prime number, p, at least one of the integers is divisible by p. Cor. 2. // neither of two integers be divisible by a prime num- ber, p, their product is not divisible by p. Cor. 3. // the product of two integers, a and b, be divisible by "an integer c and neither a nor b be divisible by c, then c is a composite number. Theorem i. Every rational integer can be represented in one and only one way as the product of prime numbers. Let m be a rational integer. If m be a prime, the theorem is evident. Let m be a composite number ; m then has some divisor, a, other than ±mor ± i. Either a is a prime or it has some divisor, b, other than ±oor±i. If & be not prime, it has some divisor, c, other than ± i and ± b. Proceeding in this manner, we must at last arrive at a prime number, for the integers of the series a, b, c, •• •, decrease in absolute value, and since there are only a finite number of integers smaller in absolute value than m, the series can have only a finite number of terms, the last of which will be a prime number ; for otherwise the series could be extended. Let this prime be p x . By §3, I, p x is a factor of m and we have m = p 1 m 1 . If m x be a prime, the resolution of m into its prime factors is complete. If ;;z x be a composite number, it contains a prime factor, p 2 , and we have m 1 = p 2 m 2 , or m = p 1 p 2 m 2 . If m 2 be not a prime, we can proceed as before until we have resolved m into factors', all of which are primes. That there will be only a finite number of these factors is evident from the fact 20 THE RATIONAL REALM INTEGERS. that the integers of the series, m, m 1} m 2 , • • •, decrease in absolute value and hence must be finite in number. We have now shown that the representation of an integer as a product of a finite number of primes is always possible. It re- mains to be proved that this representation is unique, regarding representations as identical, which differ only by the substitution for a prime of its associate. Let m = p x p 2 p z --pr = q 1 q 2 q s ~-q, be two representations of m as a product of prime numbers. Since the product q 1 q 2 "-q 8 is divisible by p lt at least one of its factors, say q lf must be divisible by p x . But q x has only the divisors ± q x and ± i. Hence q 1 =±p 1 ; that is, q x is asso- ciated with p v Then follows Pip*"' pr=±q 2 qz--- q*- In the same manner we can show that some factor of the product q<&z "*<7« is associated with p 2 , and proceeding similarly we can show that for each prime that occurs once or oftener as a factor of the product, p x p 2 p z ■ • • p r , there occurs at least as often an asso- ciated prime in the product q x q 2 q z • • ■ q 8 - In like manner, we can show that for each prime which occurs once or oftener as a factor of the product q x q 2 q 3 •••#«, there occurs at least as often an asso- ciated prime in the product p x p 2 p z •" pr> Hence the two repre- sentations are identical. We can simplify the representation of a composite number as the product of its prime factors by express- ing the product of associated prime factors as a power of one of them. Thus, if of the prime factors of m, e x are associated with p lf e 2 with p 2 , -",e r with p r , we can write m = ± p! ei p 2 e2 '" P Cr r • Cor. i. If a and b be prime to each other and c be divisible by both a and b, then c is divisible by their product. Cor. 2. // a and b be each prime to c, then ab is prime to c. Cor. 3. // a be prime to c and ab be divisible by c, b is divis- ible by c. THE RATIONAL REALM INTEGERS. 21 Theorem 2. If U (*) = a ^ + a ^ m ~ x + • - • + a ™> f 2 (x) = & ^» + b x X^ + . • . + bn, be any two integral functions of x, whose coefficients are rational integers, having in each case no common divisor, then the coeffi- cients of the product of these functions are rational integers without a common divisor. If the coefficients c , c x , • • •, c m+ n of f(x) have a common divisor other than ± I, there must be at least one prime number which divides all of them. Let p be such a prime and suppose that p divides cto, a x , • - -, a r _!, but not a r , and b , b lt • • •, b 8 _ x , but not b 8y where in accordance with our original assumption that the coeffi- cients of f x (x) and f 2 (x) have no common divisors, o^rgfw and O&sgn. We have now c r+8 = a r b 8 + a r _ x b 8 + x + Or. 2 b 8+2 + ' • ' + a r+ib 8 _ x + a r+2 b 8 _ 2 + • • •. It is evident that c r+8 is not divisible by p, for a r b 8 is not divisible by p, neither a r nor b 8 being divisible by p, while all the remain- ing terms are divisible by p, since each of these terms contains as a factor some one of the coefficients ao,a x ,~',a r - 1 ,b ,b 1 ,---,b 8 _ 1 , which are all divisible by p. Hence the coefficients of f(x) have no common divisor. Theorem 3. // f x O) == *• + a x x^ + . . . + a m , f 2 (x)=x" + b x x^ + -- + b n be two rational integral functions of x, the coefficients of the 22 THE RATIONAL REALM INTEGERS. highest powers of x in each case being i, and the remaining coeffi- cients rational numbers, the coefficients, c lt c 2 , • • •, c m+n of their product f(x) =/, (x) • f 2 (x) =x~* + c x x***-* + • • • + c m+n cannot all be rational integers unless all of the coefficients a lf a 2 , •••, a m , b lt b 2 , '",b n are rational integers. 1 Let a and b be the least common denominators of the coeffi- cients of f t (x) and f 2 (x) respectively. Then each of the func- tions a f 1 (x) and b f 2 (x) has rational integral coefficients without a common divisor. If now the coefficients c lt c 2 , • • •, c m+n are to be integers, the coefficients of the product, a<A>AO) • /.(*) =«A/(*), must all be divisible by a b . But by Th. 2 this is impossible unless a =i, b = i ; that is, a x , a 2 , • • • , a m , b , b x , • • •, b n are integers. Theorem 4. A necessary as well as sufficient condition that an algebraic number a shall be an algebraic integer is that the coeffi- cients of the single rational equation of lowest degree of the form f x (x) =x l +a 1 x l ~ 1 + ••• + a t = o, 1) ivhich it satisfies, shall be rational integers. If a satisfy an equation f 2 (x) = x m + M* w_1 + ' • ' + b m = o, of degree higher than the /th whose coefficients are rational num- bers, then by Chap. I, Th. 1, where f 3 (x) is a rational integral function of x with rational coefficients, the coefficient of its term of highest degree being 1. But by Th. 3 the coefficients of f 2 (x) cannot all be rational in- tegers unless the coefficients of f x (x) are all rational integers. Hence the theorem. 1 Gauss : Disq. Arith., Art. 42, Works, Vol. I. THE RATIONAL REALM INTEGERS. 23 We see therefore that the system of rational integers and that of the integers of R are coextensive, and hence that all that has been said in the preceding pages concerning rational integers may now be looked upon as applying to the integers of R. Hereafter the terms rational integers and integers of R will be used inter- changeably. It is seen from the above theorem that the equation of lowest degree 6i the form i) satisfied by an algebraic number, determines not only the degree of the number, but whether it is or is not an algebraic integer. After having proved the unique factorization theorem we could have shown that no rational fraction alb, where a and b are prime to each other and fr=j=± i, can satisfy an equation of the form i) whose coef- ficients are rational integers and hence that the only integers of R are the rational integers, but it has seemed better to treat the question in the general manner we have used above. § 8. Divisors of an Integer. We can now exhibit in a very convenient form all divisors of any given integer, m, and deduce therefrom simple expressions for the number and the sum of these divisors. Let m be written in the form m = ± p x ei p 2 e2 ' ' ' p r er , where p lf p 2 , '-,pr are the different prime factors of m. If d be a divisor of m, it can contain as factors only those primes which occur in rn, but each of these primes can occur in d to any power not greater than that to which it occurs in m; that is, every divisor of m must have the form d=±p i mipm 2 ... p r mr } where o^mig^.; % =5= 1, 2, •• *, r, and each of the integers obtained by giving these different values to m lf m 2f "',m r is a divisor of m. We can now easily obtain an expression for the number, AT, of the different divisors of m, associated divisors being considered as identical. Since there are e 1 -f- 1, e 2 -\-i, • ••, e r -\-i possible values for m 1 , m 2 , •••, m r respectively, there are (^i + i)(^ 2 + I )""(^r+i) different sets of values of m lf m 2 , •••, m r and each of these sets gives a dif- 24 THE RATIONAL REALM INTEGERS. ferent divisor of m. Moreover, these sets of values of m 19 m 2 , • • •, m r give all the different divisors of m, whence we have N=(e 1 +i)(e 2 + i)-.-(e r + i). We can find similarly an expression for the sum, S, of the dif- ferent positive divisors of m. On expanding the product '~(I+Pr + Pr 2 +---Pr"), we obtain a series, all of whose terms are positive divisors of m, each positive divisor of m occurring once and but once. The sum of this series is therefore S. Hence S=(i+p 1 + p 1 2 -i---'+p^)(i+p 2 + p 2 2 +"-p^) ~'(I+Pr + Pr 2 +--Pr er ) _ A ei+1 - i . A* +1 - J . . . A er+1 - * A - i' ' A- 1 A- i Ex. Let m = 6o = 2 2 -3-5. We have #= (2+ 1) (1 + 1) (1 + 1) — 12, and Saz^ii ^ZZl . 5jhI =7 . 4 . 6 _ l68 2—1 3 — 1 5 — 1 results which are easily substantiated bv observing that the positive divisors of 60 are J, 2, 3, 4, 5, 6, 10, 12, ic. 20, 30 and 60. We observe that N depends only upon the exponents of the powers to which the different prime factors appear in m, while S depends also upon the absolute values of these primes. We have defined (§3) a common divisor and a common mul- tiple of two or more integers. The representation of an integer as a product of its different prime factors leads us to convenient expressions for the common divisors and common multiples of a system of integers. Let m^, Wv,, •••,mic be any system of integers and suppose each integer of this system expressed as a product of powers of its different prime factors. Let p lt p 2 , • • •, p r be the different prime THE RATIONAL REALM INTEGERS. 2$ factors of m x ,m 2 , --',^1^', l x ,l 2 , --,l r , the exponents of the lowest powers, and g lt g 2 , • • •, g r , the exponents of the highest powers to which p lt p 2 , -",pr occur in any of these integers. Remembering now that every common divisor of m lt m 2 , • • •, m*, can contain as a factor a prime, pi, to a power not higher than the lowest to which pi occurs in any of the integers m x ,m 2 , ~',mk, we see that every common divisor of m x , m 2 , • • •, w fc , has the form where o^di^h; i=i,2,--,r. When d lt d 2t '" $ d r have their greatest possible values, that is, Kt h> ' ' '$ k, the divisor so obtained, is evidently the greatest com- mon divisor of «»!,%•",% Denoting the greatest common divisor of m x ,m 2 , ~',mu, by g, we have therefore g = Pi h P 2 l2 --pr lr . Likewise since every common multiple of m^, m 2 , • • •, m k , must contain as a factor a prime, pi, at least to the highest power to which pi occurs in any one of the integers % m 2 , • • •, m*, we see that every common multiple of m lf m 2 , • • •, m* has the form apSW--- p r nr , where **$gi, i=i,2,-- -, r, and a is any integer. When n x ,n 2 ,---,n r have their least possible values, that is, gi>g2>'">gr, and a is a unit, the multiple obtained is the least common multiple of m lt m*>, • * •, m*. Denoting the least common multiple of m x , m 2 , -•-, m* by I, we have therefore l = p^p 2 9 *---pr 9r . We observe that just as the common divisors of a system of in- tegers are the divisors of the greatest common divisor of the sys- tem, so every common multiple of all the integers of the system is a multiple of their least common multiple. When two or more of the integers m lt m 2 , • • •, mjc are prime -to each other, the greatest common divisor of the system is evidently a unit, and when the integers m x , m 2 , • ■ •, m& are prime each to each their least common 26 THE RATIONAL REALM INTEGERS. multiple is their product, m x m 2 • • • m k . If an integer be divisible by each one of a system of integers m lf m 2> • ■ •, w&, it is divisible by their least common multiple. If we have two integers = P x *Pf*' ' • p r a % b = p^pj* • • • />r 6r , and g ■ = p 1 hp 2 h ... p r ir f l = pjhpp... p r ffr be respectively their greatest common divisor and least common multiple, it is evident that li + gi = <h + K h + g2 = ^ + b 2 r"Jr +gr = a r + b r , and hence that gl= ab ; that is, the product of two integers is equal to the product of their greatest common divisor and least common multiple; for example 12 -30 = 6-60. The representation of an integer m as a product of powers of its different prime factors gives us also a criterion for determin- ing whether in is or is not the &th power of an integer. Let m = ± p x ei p 2 e2 ' ' ' pr er . By putting m = n k , we see immediately that the necessary and sufficient condition that m shall be the &th power of an integer is, if k be odd, that e lf e 2 ,--,e r shall be divisible by k, while if k be even there is the further condition that m shall be positive. § 9. Determination of the Highest Power of a Prime, p, by which m ! is divisible. The method employed consists in counting, successively, those : nte£ers of this product which are divisible by p, p 2 , p 3 , etc., respectively. Remembering that those integers which are divis- ible by p l have already been counted i — 1 times, as among those divisible by p, p 2 , •••, p l ~ x , the sum of these enumerations is seen to be the exponent of thi desired power of p. Denote this expo- nent by e. Since e will have the same value for — p as for p, we can without loss of generality assume p positive. Let [a/b] denote the greatest integer contained in the fraction a/b, where a and b are both positive; in particular [a/b] is o THE RATIONAL REALM — INTEGERS. 2 J when a is less than b. Put [m/p] = m lf [m/p 2 ] = m 2 , • • •, [m/p*] =«*, • • • . There are in the product i»/=i*2'3 ••• m, the m, integers, />, 2/>, Zp,-"^xP, 1 1 divisible by /^ and wz/ is therefore certainly divisible by ^"i; that is, e < w x . In like manner there are in ml the m 2 integers p\ 2 p 2 ,--;m 2 p 2 2) divisible by p 2 . We have counted these integers once already among the integers i), but since they each contain p twice as a factor, and there are m 2 of them, we must add m 2 to the exponent of the power of p which is known to divide ml. Hence ml is certainly divisible by p m ^ m * ; that is, e <£ m i + m 2- Likewise there m 3 integers of ml divisible by p 3 , each of which has been counted twice already. Hence e < *>h + ™ 2 + "h- Continuing this process we arrive finally at a fraction m/p k > which is less than i, and hence <*-[$] -a The highest power of p by which ml is divisible is therefore p mi Hn 2+ --+m k - 1} w hose exponent e is [m/p] + [m/p 2 ] -| + [*•/£***]. If p > m, then w 1 = o, and hence e = o. Ex. Let m = ioo, and p = .3 ; then W ,= [W]= 3, « 4 =[W]= I, w»=[Hf]= o, and ^ = 33 + n+3-J-i = 48. It is easily shown that L;l m- 28 THE RATIONAL REALM INTEGERS. and hence -M Using this fact in the example just given we have Wi= [ x §-] = 33> m * = [¥] = ii, w 3 = [-V-1 =3, ™*= [f ] = Xj «»= CH = o. m/ § io. The Quotient 7 , , where m = a-\-b -\ + fc, is a/&/ ••• &/ an Integer. 1 This quotient will be recognized as the so-called multinomial coefficient ; that is, the coefficient of x x a x* • • • x r k in the expansion of (x 1 + x 2 -f- • • • x r ) m . When r = 2, and m = a-\-b, we have the binomial coefficient ; that is, the coefficient of x x a x 2 m - a in the expansion of {x x -\- x 2 ) m . This theorem is easily proved by means of that of the last sec- tion. To show that . T ' . , i) is an integer it is necessary and sufficient to show that every prime, p, is contained to as high a power in the numerator as in the denominator. Let e, a x , b lt ■ • •, k lt be the exponents of the highest powers to which p is contained in ml, al, b!,'-,k!, respect- ively. We must show that **«k 4- &! + "•■+"** Since m= a + b -f- * • * + k, . mad k it follows that — = - H 1 4- — , -«- [7]i4 + L-]*--[i] [?3*[?]*&+~[?} ;;/ The truth of this theorem is at once evident since —m ; - is the alol • • • k! number of permutations of m things a, b, • • ; k of which are alike. THE RATIONAL REALM INTEGERS. 20, Hence, by addition, r m~\ + ••• [7H7V Aj] * ]+[ f ]+ +L : ] + [f] + [?] + +[?] •■-[i] + [?i + - + [?] + - Hence £ ^ Oj_ + & x + • • • + k x . Therefore p is contained to at least as high a power in the numerator of I ) as in the denominator. But p was any prime ; therefore I ) is an integer. From this it follows that the product of any m successive posi- tive integers is divisible by ml For (a-f-i) (g+ 2 ) ' ' •(fl+ftQ _ a/(a+i)(a+2)-"(o+w) _ (g+ w )' ml aim! at ml which is an integer. From this and the fact that o is included among m successive integers which are not all positive or all neg- ative, it follows that the product of any m successive integers is divisible by ml 30 the rational realm integers. Examples. 1 ' i. The sum of two odd squares can not be a square. , 2. Every integer of the form 4» — I has an odd number of factors of the form 4ft — 1. 3. Every prime greater than 5 has the form yym'-^n where w=i, 7, 11, 13, 17, 19, 23 or 29J 4. The square of every prime greater than 3 is of the form 24m + 1, and the square of every integer which is not divisible by 2 or 3 is of the same form. 5. If n differ from the two successive squares between which it lies by x and y respectively, prove that n — xy is a square. 6. The cube of every rational integer is the difference of the squares of two rational integers. 7. Any uneven cube, n 3 , is the sum of n consecutive uneven integers, of which n 2 is the middle one. 8. Show that x 3 — x is divisible by 6 if x be any integer. 9. Show that x 4 — 4X 3 + S x2 — 2X 1S divisible by 12 if x be any integer. {o. Show that x 4m -f- x 2m + 1 never represents a prime number if x be any integer other than 1. 1 1. r Show that (mn) ! is divisible by (m!) n n! 12. Show that (2w) !{2n) ! is divisible by ml nl (m + m) / 13. What is the least multiplier that will convert 945 into a complete square? 14. Find the number of the divisors of 2160 and their sum. 15. Find a number of the form 2 n '3-a (a being prime) which shall be equal to half the sum of its divisors (itself excluded). 1 See Chrystal; Algebra, Part II, pp. 506, 518 and 526 for other examples, also C. Smith, Algebra, and Hall and Knight, Higher Algebra. CHAPTER III. The Rational Realm. congruences. § i. Definition. Elementary Theorems. // the difference of two integers, a and b, be divisible by an integer m, a and b are said to be congruent to each other with respect to the modulus m. This relation is expressed by writing a^=b, modm. 1 Similarly, if the difference of a and b be not divisible by m, we say that a and b are incongruent to each other, with respect to the modulus m, and write a^=b, mod m. Ex. We say that 21 is congruent to 15 with respect to the modulus 3, since 21 — 15 is divisible by 3. In the above notation this fact is ex- pressed by writing 21 == 15, mod 3. We can express the fact that a is congruent to b by writirlg a — b = km, or a = b + km, where k is an integer, but the notation a==b, mod m, which is due to Gauss, has the great advantage of placing in evidence the analogy between congruences and equations ; and we shall see that most of the transformations to which equations can be sub- jected are also applicable to congruences. H. J. S. Smith says : " It will be seen that the definition of a congruence involves only one of the most elementary arithmetical conceptions, — that of the divisibility of one number by another. But it expresses that conception in a form so suggestive of anal- ogies with other parts of analysis, so easily available in calcula- tion and so fertile in new results that its introduction into arith- 1 The author has adopted a slight variation of Gauss's notation, a = b (mod m), due, he believes, to H. J. S. Smith. 31 32 THE RATIONAL REALM INTEGERS. metic (by Gauss) has proved a most important contribution to the progress of the science." We have as direct consequences of the de'finition of congruences the following: i. If a = b, modw, i) and b = c, modw/ 2) then a = c, modm; for, from 1) and 2), we have respectively a — b = km, b — c = k x m, where k and k x are integers, and by addition a — c= (k -\- k x )m', that is, a = c, mod m. It is now evident that we can divide all integers into classes with respect to a given modulus, if we put into the same class those and only those integers which are congruent to each other with respect to this modulus. We ask: How many such classes will there be for any given modulus m? Any integer, a, can be written in the form a = km + r, where k is an integer and r is one of the integers o, 1,2,3, •••,|m|— 1. But a is congruent to r, mod m, and if we give k all integral values from — 00 to +00, the resulting values of a will be a series of integers, all of which are congruent to r, and hence by i to each other with respect to the modulus m. By putting for r the I m I different values o, 1, 2, 3, •••, | m \ — 1, we shall get | m | classes and every integer is seen to fall into one or the other of these classes. An integer can not be in two different classes, for then we should have a = km + r = k x m + r u OF i££LlFOBl^ THE RATIONAL REALM CONGRUENCES. 33 where r= ¥ r i, - which gives ( k — k 1 )m = r 1 — r. Since the first member of this equation is divisible by m, the second member must be divisible by m also, but since r and r t are both positive and less than | m |, we have \r — r x \ < \m\, and hence r — r x is not divisible by m, unless r — r x = o\ that is, r — r x and hence k = k 1 . There are therefore exactly | m | incongruent number classes with respect to the modulus m, each integer being in one and but one of the classes. The absolute value of an integer, m, may now be defined as the number of incongruent number classes with respect to the mod- ulus m. This definition brings out clearly a reason for the introduction of the absolute value of an integer ; that is, to express the result of an enumeration as a function of an integer. In all theorems relating to congruences we shall think of the entire system of rational numbers as divided into such classes, with respect to some given modulus ; and whatever is true of any individual integer with respect to this modulus will be true of the entire class to which it belongs. We shall thus deal rather with the classes than with the individuals in them and it will only be necessary to have a representative of each class. Such a system of | m | representative integers, each integer being chosen arbitrarily from the class to which it belongs, is called a complete system of incongruent numbers, or a complete residue system, with respect to the modulus m. The latter designation is derived from an extension of the ordi- nary idea of the remainder, which holds when the system chosen is o, 1,2, ", \m\ — 1, by calling either one of any two integers, which are congruent to each other with respect to the modulus m, a remainder or residue of the other with respect to m. Any I m | consecutive integers evidently form a complete resi- due system with respect to the modulus m. The most useful systems are, first, that composed of the small- 3 34 THE RATIONAL REALM — CONGRUENCES. est possible positive residues o,i,2,-",\m\ — i, and second, that composed of the residues of smallest possible absolute value, the latter being, when m is odd and | m | =2» + i, — n,— (n — i),-..,_ 1,0, i, •••,« — i,«; and, when m is even and \m\=2n — (n — i), •••, — 1,0, 1, "' 9 n — 1,«, the two residues n and — n being congruent to each other, mod m. Ex. If m = ii, each of the systems o, i, 2, 3, 4, s, 6, 7, 8, 9, io; — 5, —4, —3, —2, -M, o, i, 2, 3, 4, s; 50, —15, —25, 20, 32, 22, —io, 13, —19, 4, 16 is a complete residue system, mod II. ii. Addition and subtraction of congruences. If a 1 = b 1 , mod m, 3) and a 2 z=zb 2 , modm, 4) then a, ± o 2 aa fr x ± b 2 , mod m ; for we have from 3) and 4), respectively, a i — &i = &i w j #2 — & 2 =3 k 2 m f whence {a x ± a 2 ) — (& x ± & 2 ) — (A ± & 2 ) w ; that is, (a ± ±a 2 )^=b 1 ± b 2 , mod m. iii. Multiplication by an integer. If a = b, mod ni } 5) then ac^bc, modw; for from 5) we have (a — b)=km; whence ac — be = kem ; that is, ac^bc, modw. iv. Multiplication of congruences. If a x ^b u modw, 6) I THE RATIONAL REALM CONGRUENCES. 35 and a 2 = b 2 , modm 7) then a 1 a 2 ^bjb 2i modm; for from 6) we have by iii a x a 2 = b x a 2 , modm; and similarly, from 7) b r a 2 ^b x b 2 , modm, whence by i a x a 2 = b x b 2 , mod m. From this it follows, evidently, that if a = &, modm, then (& = &, modm, where k is any positive integer. v. If f(x) be a polynomial in x with integral coefficients; that is, f( x )= %*" + 0i* n_1 + • • • + On, and if r^r x , modm, then f(r)^=f(r x ), modm, 8) for from r = fi, modm it follows by iv and iii that a . r n-i == tfif^*-*, mod m, i = o, 1 , 2, • • • , w, and by addition we obtain 8). It may be shown similarly that if f(x lf x 2 , '-',x n ) be a poly- nomial in x lf x 2 , --,Xn with rational integral coefficients, and if a 2 — ^2 L modm, On = i n J then f(a 1 ,a 2 ,'-,a n )=f(b 1 ,b 2 ,--,b n ), modm. Ex. Let f(x)=2x 3 — x* + s; since — 3 ebeII, mod 7, we have /( — 3) =/(n), mod7; that is, — 58 = 2546, mod 7. 36 THE RATIONAL REALM — CONGRUENCES. vi. Removal of a common factor. We have seen in III that we can multiply both members of a congruence by any integer, without affecting the validity of the congruence ; the converse of this, however, is not in general true. Thus we have 8 aa 14, mod 6, but , 4 4 s 7, mod 6. To consider this question in general, let a==&, modw> be a congruence in which a and b are both divisible by k ; that is, a = a x k and b = b x k. where a x and b x are integers. Then from a x k = b x k, mod w, it does not necessarily follow that a x = b lt mod m ; for that a x — b x shall be divisible by m is not a necessary conse- quence of k(a x — b x ) being divisible by m, unless k be prime to m, and all we can say in general is that a x — b x is divisible by those factors of m which are not contained in k ; that is, by m/d, where d is the greatest common divisor of k and m. Hence from a x k = b x k, mod m, it follows in general only that a x m b v mod -^, 9) where d is the greatest common divisor of k and m. If k be prime to m, d is 1, and hence from 9) we have a x ^==b x , modm. Ex. From 8^14, mod 6, it follows that 4^7, mod 3; but from 5 = 35, mod 6, it follows that 1 == 7, mod 6. THE RATIONAL REALM — CONGRUENCES. 37 vii. // a = &, mod m, and d be a divisor of m, then a^=b, mode?; for since a — b is divisible by m it is divisible by d. viii. If a=b with respect to each of the moduli m v m 2 , •••, m n , then a = b, mod I, where I is the least common multiple of m lf m 2) --,m n \ for since a — b is divisible by each of the integers m x , m 2 , ••, m n , it is divis- ible by their least common multiple. An important special case of this is when m lt m 2 , ••yw* are prime each to each, / being then their product. ix. All integers belonging to the same residue class have with the modulus the same greatest common divisor; for if a = &, modw, then a — b = km; and any integer that divides a and m must also divide b, and any integer that divides b and m must also divide a. Therefore the greatest common divisor of a and m is identical with the greatest common divisor of b and m. In particular // any integer of a residue class be prime to the modulus m, then all the integers of this class are prime to m. §2. The Function <f>(m). By ^(m) 1 we denote the number of integers of a complete residue system, mod m, which are prime to m. Such a system of integers is called a reduced residue system, or a reduced system of incongruent numbers, mod m. That the number of integers in such a system is independent of the complete residue system chosen is obvious from § i, ix. We can therefore calculate <£(w) for a particular value of m by writing down any complete residue system, mod m, and removing those integers of this system that are not prime to m. The number of those remaining is evidently 4>(m). 1 The symbol is due to Gauss : Disq. Arith., § 38, Works, Vol. I. Euler first gave a general expression for <t>{m) : Comm. Arith., I, p. 274. 38 THE RATIONAL REALM CONGRUENCES. Thus for m = — 10, take as a complete residue system — 10, — 19, 2, — 7, — 16, 5, 16, 17, 18, — 1. Striking out the integers — 10,2, — 16,5,16,18, that are not prime to — 10, we have left the four integers — 19, — 7, 17, — 1, that constitute a reduced residue system, mod — 10. Hence <j>( — 10) = 4. As a second example, let m = 7. A complete residue system, mod 7, is 0,1,2,3,4,5,6, and we see that <f>(y)=6. The last example leads Us at once to a general expression for <t>(p), when p is a prime ; for the integers o, I, •••, | p | — 1 con- stitute a complete residue system, mod p, and are, with the excep- tion of o, all prime to p, whence it is evident that <t>(P)=-\P\ — 1. It should be observed that, since the units are regarded as prime to themselves, 4>(±i)=i. The first method, which we shall employ to obtain a general expression for <f>(m) in terms of m, is exactly similar to that em- ployed in the examples just given ; that is, we write down a com- plete residue system, mod m, remove those integers of this system which have a common divisor with m, and count those remaining, their number being <f>(m). The general expression for <f>(m), where m is any integer, is given by the following theorem : Theorem i. // p lt p 2 > '-,p r be the different positive prime fac- tors of m, and <f>(m) denote the number of integers of a complete residue system, mod m, that are prime to m, then ^(m) = |m|(i-f)(i-^)-(i-^-). Pi P 2 Pr Since, evidently, <f>(—m)=<f>(m), we can without loss of generality assume m positive. THE RATIONAL REALM — CONGRUENCES. 39 Let tn = p 1 ei p 2 e2 '- pr er , where p x ,p 2 , '",pr are the different positive prime factors of m. Take as a complete residue system, mod m, i,2,3,4,---,m S) Our task is to remove from the system S those integers which are divisible by one or more of the primes p lt p 2 , '",pr, and to count the integers left. We shall first remove those divisible by p lt namely the m/p 1 integers m Pi,2px,2>Px>'->irPx> Vx Removing these from S there remains a system, S lf consisting of m — tn/p lt —rn(i — l/P % ) , integers, none of which is divis- ible by p x . From this system S x we must now remove those integers that are divisible by p 2 ; that is, those integers of 5 which are not divis- ible by p x but are divisible by p 2 . The integers of S which are divisible by p 2 are the m/p 2 integers m p2> 2 p2,3p2,'--,rp 2 ,---, — p 2 , A and the necessary and sufficient condition that any one, rp 2 , of these integers be also divisible by p lt is that the coefficient, r, of p 2 shall be divisible by f v The number of the integers, which are to be removed from the system S\ on account of their divisibility by p 2 , is therefore the same as the number of the integers m 1,2,3, ~, ~> A which are not divisible by p lt and this is, since m/p 2 is divisible by p x , exactly as in the first step of this proof j\ \ m t-$ 40 THE RATIONAL REALM — CONGRUENCES. There remains then of S a system, S 2 , of integers, none of which is divisible by p r or p 2 . We are now led to conclude by induction that the number of the integers of S, which are divisible by none of the r primes p lt p 2 , •-, p r is m ('4)(-i)-(-7.) m To prove that this is correct, it is only necessary, since we know that it holds for r = 2, to show that, if it holds for r = i, it holds for r = i-\- i. Assume then that, having removed from 5* the integers divisible by one or more of the i primes p lt p 2 , '",pi, there is left a system (-*)(-*)• •■(-*) | integers. To obtain the number of integers of S that are divisible by none of the primes p 1} p 2 , '",pi+i, we must remove from Si those integers which are divisible by pi +1 and count those remaining. The integers of Si that are divisible by pi +1 are the same as the integers of 5 that are divisible by p t+1 but are divisible by none of the primes p 1} p 2 , •••, pi. The integers of S that are divisible by p i+1 are m Pi+v 2 A+i, • ' '» r A+v ' ' •• J-Pi+v Pi+\ and the necessary and sufficient condition that any one rp i+1 of these integers shall be divisible by none of the primes p lf p 2t '",pi is that the coefficient, r, of pi +1 be divisible by none of these primes. The number of integers to be removed from St coincides there- fore with the number of the integers m A+i THE RATIONAL REALM CONGRUENCES. 4 1 that are divisible by none of the primes p lf -",pi' By formula i), whose correctness has been assumed, this number is t( l -.k)( l -d-( l - l p} m A Subtracting this number from i) we get ■■i'-k)-(-j)(-k){-k)(-^ an expression identical in form with i), as the number of the integers oi 5* which are divisible by none of the primes Pi,p2>'--,pi,pi + i- But we have proved the correctness of I ) when i = 2, hence the theorem holds when t=3, and similarly when i = r. If m be any integer, positive or negative, and p lt p 2 , •■■ J p r be its different prime factors, positive or negative, we have as an absolutely general expression for <f>(m) ^) = IH( I - f ^)-( I -^ r ). Making use of the representation of m as a product of powers of its different prime factors, we obtain another expression for <f>(m) ; that is, <t>(m) = (\p i \--l)\p i \e^ -- (\p r \ — l)\p r \er-K If m be a power of a single prime as p e , we have <t>(±p') = (\p\-i)\p\ e -\ and, in particular, when e=i, <t>(P) = \p\ — i- Ex. Let m = 60 = 2 2 • 3 • 5. We have 0(60) = 60(1 — i) (1 — *) (1 — |) 42 THE RATIONAL REALM CONGRUENCES. a result seen to be true when we write down the complete residue system, mod 60, 1, 2, • • •, 60. For when we remove those integers which are not prime to 60, there are left the integers 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, in number 16. We observe that <f>(m) is an even number except when m = ± 1, or ±2; for if m=±2 e , we have <f>(± 2 e ) =2 e ~ 1 , which is an even number when e > 1, and if m contain an odd prime factor p lt then from 2) it is evident that <f>(m) contains the even number I p I — 1 as a factor and hence is an even number. This may be proved independently of the formula. 1 The above proof, which is the one usually given for this theorem, has been used here on account of its great simplicity. It does not, however, admit of extension to the higher realms in Jhe form here given, since a property of rational numbers has been made use of which has no analogue in the case of algebraic numbers of a higher degree. We therefore give below a proof depending upon the same principles as the above but so formu- lated that it is at once capable of extension to a realm of any degree. 2 In giving these two forms we hope to make clear to the reader some of those conditions which must be satisfied by the form of proof of a theorem regarding rational integers in order that, should the theorem be found to hold for the integers of any algebraic number realm, the same form of proof can be used for it in the general case. The proof of the general theorem (Th. 1) depends directly upon the following simple theorem: Theorem 2. // a=bc, where b and c are any integers, there are in a complete residue system, mod a, exactly \c\, = |a/&|, numbers that are divisible by b. Since by §1, ix, if the theorem be true for any particular residue system, mod a, it is true for all, we shall construct |c| numbers which are divisible by b and incongruent each to each, mod a, and shall then show that no other number of a complete 1 Cahen : p. 33. 2 See p. 44. THE RATIONAL REALM — CONGRUENCES. 43 residue system, mod a, of which these numbers are a portion, can be divisible by b. Let c lf c 2 ,"-,c Cf 2) be any complete residue system, mod c. The integers bc lf bc 2 ,--,bcc 3) are incongruent, mod a, for if ben ass bci, mod a, then c h = Ci, mode, which is impossible. Moreover, every integer, bd, divisible by b is congruent, mod a, to some one of the numbers 3), for d is congruent, mod c, to some one, say Ci, of the integers 2), and from d = Ci, mode, it follows that bd = bci, mod a, and bd is one of the integers 3)! Hence the integers 3) comprise all those integers of a complete residue system, mod a, of which they are a portion, that are divis- ible by b. They are | c | in number and the theorem is therefore proved. If we select the particular residue system 1, 2, •••, \m\, and observe that the integers of this system, that are divisible by b, are, considering b positive, b, 2b, b, the truth of the theorem is at once evident. The form of proof used above has, however, been chosen on account of its immediate adaptability to the higher realms. From the above theorem we obtain at once the following : Theorem 3. If p be any prime There are in a complete residue system, mod p e , exactly | p e /p | numbers that are divisible by p and therefore \ p e \ — | p e /p \ that are prime to p. Hence the theorem. 44 THE RATIONAL REALM CONGRUENCES. We shall now prove again Theorem i, placing no restriction upon either m or its prime factors as to sign. Theorem i. If p u p 2 , • • •, p r be the different prime factors of m, and <j>(m) denote the number of integers of a complete residue system, mod m, that are prime to m, then Second Proof. 1 Denote by S a complete residue system, mod m, and let \m\ . _. | m ; J \m\ \m\ \m\ \m\ S. = rrrrr, + rrrrn + • ■ ■ + AllAl lAllAl lA-.llAl' 5 = AllAh--|Al' Consider now the sum N=\m\-S 1 J r S :! --- + (-iySr. Making use of Theorem 2, we see that an integer of S, which is divisible by i of the />'s but not by * + 1 of them, is counted once in | m |, % times in S lf i(i — i)/i-2 in S 2 , -•-, and finally once in Si. Hence this integer contributes to N the number i-i+ i ± i ^-- + (-iy=(i-iy=o. Therefore every integer of S that is not prime to m contributes o to N, while every integer oi S that is prime to m contributes 1 to N, since it is counted once in \m\ and is not counted in S ly S 2 , •••, S r . Hence N is the number of those integers of 5* which are prime to m; that is, N = <f>(m). 1 Mathews: §7. THE RATIONAL REALM — CONGRUENCES. 45 Therefore + («) = |m|— S t + S, + (-iys r = ]Ml ( l -\k\)( l -\k\)-{ l -\k\} § 3. The Product Theorem for the ^-Function. Theorem 4. // m = w 1 w 2 , where m x and m 2 are prime to each other, then <f>(m) = cf>(m 1 )cf>(m 2 ). Let m 1 = ± p 1 ei p 2 e2 • • ' pr er , and m 2 = ± q x fl qj* ' ' ■ fA where p lf p 2 ,-', pr, q lf q 2 r",q 8 are different primes. Then m = ± p t * • • • p r er q x fl ■ ■ ■ #/', and ^«) = l,«l( I - f i i] )...( I -^ | )(.- | ^)...( I -^) =w(i-^ ] )-(-^)i- 2 i( i -^ i )-(-^) Ex. Since 60 =4-15, and 4 is prime to 15, we have 0(6o) =0(4)0(15) =2-8=16 The above result can evidently be extended to a product of any number of factors, which are prime each to each; that is, if m = m 1 m 2 ■ • • m r , where m lt m 2 , • • • m r are prime each to each, then <f>(m) =<f>(m 1 )(f>(m 2 ) ••• <f>(m r ). This theorem is useful in the calculation of <f>(m). Ex. Since 315 = 3 2 • 5 • 7, we have 0(315) = 0( 3 2 )0(5)0(7) = 6-4-6 = 144. This property of the function <f>(m) can be derived without the use of Theorem 1. This having been done and having shown that ♦eo-i'iO-lTl)' 46 THE RATIONAL REALM CONGRUENCES. we can derive the general expression for <f>(m) in terms of m. This is the method adopted by Gauss. 1 § 4. The Summation Theorem for the ^-Function. Theorem 5. // d be any divisor of m and m = nd, the num- ber of integers of a complete residue system, mod m, which have with m the greatest common divisor d is <f>(n). Since by § 1, ix, if the theorem be true for any particular resi- due system, mod m, it is true for all, we may take the system used in Theorem 2. We have shown there that the system of integers dn^dn^-'-ydnn, 1) where n lf n 2i - -,n n is a complete residue system, mod n, com- prises all those and only those integers of a complete residue sys- tem, mod m, which are divisible by d. Hence the integers of this complete residue system, mod m, which have with '^ the greatest common divisor d are those of the system 1) in which the coefficient of d is prime to n. Since n lf n 2 , -",n n is a complete residue system, mod n, the number of these integers is <}>(n) and the theorem is proved. Theorem 6. If d 17 d 2f -", d r be the different divisors of m, we have Y>(«*«) = M. The proof of this theorem follows easily from the last. Write down all the different divisors,^, d 2 , •••, d r , of the integer m, and let m = m x d x = m 2 d 2 ==••• = m r d r , observing that both 1 and m are included among the divisors of m. Separate the integers of a complete residue system, mod m, into classes in the following manner. Place in the first class those integers of the system that have with m the greatest common divisor d x ; by Theorem 5 they will be ^(wj in number. Place in the second class those integers of the system that have with m the greatest common divisor d 2 ; they will be similarly <f>(m 2 ) 1 Disq. Arith., Art. 38. Works, Vol. I. See also p. 75. THE RATIONAL REALM CONGRUENCES. 47 in number. Proceeding in this way it is evident that we shall have r classes and that each integer of the system will occur in one and but one of these classes. But the number of integers in a complete residue system, mod m, is \m\. Hence the total number of integers in these classes is \m\. Since, however, the total number of integers in the classes is also ^(mj + <f>(m 2 ) + ••• +<£(m r ), and m Xi m v '",m r are merely d lf d 2 ,'--,d r in different order, we have f>(*)' s H«l- i=l Ex. Let m = 30. The different divisors of m are 1, 2, 3, 5, 6, 10, 15, 30. We have then 0(1) +0(2) + 0(3) + 0(5) + *(6) + 0(io) + 0(15) +0(30) = 30, a result which may be verified by calculating the values of 0(i), 0(2), • • •> 0(30) » and taking their sum. We have 1 + 1+2 + 4 + 2 + 4+8 + 8 = 30. The above property of the function <f>(m) has been derived immediately from the original definition of the function, no use having been made of the expression found for <f>(m) in terms of m. It completely defines <f>(m) and from it we can derive all the properties of the function, in particular the expression for <f>(m) in terms of m. 1 We shall give now another proof of this property of <f>(nt) making use of Theorems 3 and 4. In order to bring out clearly the analogy which exists between this proof and that of the corresponding theorem in the higher realms which will be given later we shall put no restriction upon either m or its prime factors as to their sign, although so far as this proof is concerned merely with rational integers, they may evidently all be assumed positive without limiting its generality. 1 Dirichlet-Dedekind : § 138. 48 THE RATIONAL REALM CONGRUENCES. Let m=± p x ei p 2 e * • ' ' p r er where p!,p2,"',pr are different primes. Every divisor of m has the form di = ±p x ^p^--'pr fr i) where f x is one of the numbers o, I, ••• *£, f 2 is one of the numbers o, i, ••• e 2 , f r is one of the numbers o, i, ••• e r . We have by Theorem 4 ♦ (A) =+(P% H )4>{Pt U ) '"4>{Pr fr )- 2) If we let f lf f 2 ,"-,fr run through the values o, 1, • • •, e 1 ; o, 1, • • •, e 2 ; ••• ;o, 1, --',e r , respectively, we obtain from 1) all the divisors of m, and from 2) the corresponding values of <f>(di) whose sum is 1=1 We see therefore that the terms of the series obtained by multi- plying out the product P= [#(x) + *(*) + *<*■) + ••• + 4>(/> 1 e 0] - are identical with the terms of I>(rfO ; »=l r that is, P =£<£(</,). i=l But *(i) = i, <t>(p 1 ) = \p 1 \-h'-, <f>(Pi e i) = \Pi\ e >- 1 (\Pi\ — i), whence *(0 + <£(/>i) + ••• + 4>(/>i e = I />i | e S and similarly for the other factors of 3). Therefore P=\Pi\ e *\P 1 \ e *--\pr\ e '=\m\, THE RATIONAL REALM CONGRUENCES. 49 and hence ±<t>(d i ) = \m\. § 5. Discussion of Certain Functional Equations and Another Derivation of the General Expression for <j>(m). Theorem 7. // m be any integer other than ± 1, whose dif- ferent prime factors are p. lt p 2 , '"> pr, and d any divisor of m other than dz m, and if we separate all integers of the form m Pl,P 2 '"Pi no p being repeated, into two classes, I and II, putting in class I those such that m is divided by none or by the product of an even number of the p's, and in class II those such that m is divided by the product of an odd number of the p's, then exactly as many integers of the one class are divisible by d as of the other. 1 Before proving this theorem it will be well to illustrate its content by an example. Let m = 6o = 2 2 .3.5. Forming the above mentioned numbers we have the following : ^ _ . 60 60 60 , . _ Class 1 : 00, — , — , — ; that is, 00, 10, 6, 4. 2.3 2 -5 3-5 _, TT 60 60 60 60 , . Class II: — , — , — , ; that is, 30,20, 12,2. 2 3 5 2.3.5' '° If now d=io, we see that two numbers of each class are divisible by 10; that is, 60 and 10 of I, and 30 and 20 of II. We proceed to prove the theorem, observing that since we are concerned here only with questions of divisibility and since in such questions what is true of one associate of an integer is true of both of its associates, we may without limiting the generality of our proof assume m,p x , --ypr and d to be positive. Making this assumption, we see that the positive and negative terms of the developed product 1 Dirichlet-Dedekind : § 138. 4 50 THE RATIONAL REALM CONGRUENCES. VI (<-&(->)■;•('-*) ■> coincide respectively with the integers of I and II. That is, denoting by %m lt 2m 2 , respectively, the sums of the numbers of these classes, we have Let we shall first prove the theorem for the case in which C 1 = € 2 = ' ' ' == €r = I J that is, m is not divisible by a higher power than the first of any prime. Setting p x p 2 • • • p r = a, we have a ( I -7 1 )( I -7 2 )-( I -i)= (A - I) ^- I) -^- I) tm 2^ — 2tf 2> where ^a x ,^a 2 have meanings corresponding to those of ^m Xi ^m 2 . If now b be any positive divisor of a other than a, the number of the a x terms that are divisible by b is exactly equal to the num- ber of a 2 terms that are divisible by b, for, if we put a = bq 1 q 2 ••• q 8 where q lt q 2 ,'">q8 are those prime factors of a which do not divide b, then the a x terms and the a 2 terms that are divisible by b are respectively the positive and negative terms of the developed product b(q 1 — i)(q 2 —i) ••' (?•— I). 2) Moreover, since b=%=a there is at least one prime, q, that di- vides a but not b ; that is, there is at least one q. Hence there are exactly as many positive as negative terms in the developed product 2) and consequently as many of the a/s as of the a 2 , s are divisible by b. THE RATIONAL REALM CONGRUENCES. 5 I The theorem is therefore proved for the case in which m is not divisible by a higher power than the first of any prime. We proceed now to prove the theorem for the general case. Let a, a lf a 2 retain the meanings assigned above. We have m = p^p^ 1 • • ■ p r er - 1 p 1 p 2 • • • p r = na, and it is evident that the integers m lf m 2 coincide respectively with the products na lt na 2 . Now let d be any positive divisor of m other than m and let g be the greatest common divisor of the two integers d = gb, n = gc. We see that b is a divisor of a ; for ca/b is an integer since 3) ca gca na m which is an integer, and c is prime to b. From 3) it follows, since c is prime to b, that, if d = m, then c=i and b = a. Conversely, if b be equal to a, and hence be divisible by all prime factors of m, then c must be I, since it is a divisor of m but prime to b, and hence d = tfk Excluding, therefore, the case d = m, so that we have always b=$=a, there are among the integers Oj exactly as many that are divisible by b as there are among the integers a 2 . Since, moreover, the necessary and sufficient condition that an integer m ls or m 2 , where m 1 = na 1 = gca 1 , or m 2 = na 2 = gca 2 , shall be divisible by d = gb, is that a 1? or a 2 , shall be divisible by b, there are exactly as many of the integers m x divisible by d as of the integers m 2 . The theorem is therefore proved. Many interesting applications may be made of this theorem; among them are the two following : 52 THE RATIONAL REALM CONGRUENCES. Theorem 1 8. A) If f(m) and F(m) be two functions of an integer m that are connected by the relation Sf(d)=F(m), 4) where d runs through all divisors of m including m, then f(m) = ^F(m^) — SF(w 2 ), 5) where m lf m 2> run through the values defined in the last theorem. B) If f(m) and F{m) be connected by the relation Uf(d)==F(m) 6) where the product relates to the values of the function corre- sponding to all the values of d, then ' (jw)= nFky 7) To prove A it is sufficient to observe that if d be any divisor of m other than ± m, it is a divisor of exactly as many of the ■m/s as of the w 2 's (Theorem 7), and hence, when in 5) we replace the F's by their values in terms of the /'s from 4), f(d) will occur exactly as often with the plus sign as with the minus sign. Hence all terms in the second member of 5) will cancel except f(m) which occurs once only. We shall illustrate this by a numerical example. Letw=i5. We have 15(1 -i) (1 -i) = 1-3-5 + 15 = 1 + 15- (3 + 5), whence ^m x =1 + 15, and 2m 2 = 3 + 5. Also from 4) /(i)+/(3)+/(5)+/(i5)=f(i5). /(i)+/(S) =-P(5), /(0+/(3) =F(3), /(i) =F(i). 1 This theorem holds also in the case m = 1, which was excluded in Th. 7, if we assume that in this case there is only a single m h = 1, and no Iff* THE RATIONAL REALM — CONGRUENCES. 53 We have now from 5) /(i 5 )=2F(7» 1 )-SF( W , 2 ); for /(i 5 )=F(i) +F(i 5 )- [F( 3 ) +F( 5 )] = /(i)+/(0 +/(3) +/(5) +/(I5) -(/(i)+/(3)+/(i)+/(5)) = /(iS)- The proof of B is evidently exactly like that of A. It will suffice if we illustrate it by a numerical example. Let w== 15 ; we have from 6) /(i)/(3)/(S)/(i5)=^(i5), /(i)/(5) =F(5), /(i)/(3) =-F(3), /(i) =F(i). From 7) _F(i)F(i S ) _ ^(3)>"(5)"' _ /(i)-/(i)/(3)/(5)/(i5) /(iV(3)-/(0/(S) ' —/(M). From Theorem 8, A, we can easily deduce by the aid of Theorem 6 the general expression for <f>(m). From Theorem 6 we have where d runs through all divisors of m. Applying Theorem 8, we have f(m) = <£(w) and F(m) = | m |, 54 THE RATIONAL REALM CONGRUENCES. and hence W =^ I -2^ = | OT |( I -^)( I -^- | )...( I - r/ i- | ). As an example of the use of Theorem 8, B, we give the fol- lowing : Let f{m) = p, when m is a power of the prime number p, and f(m) = i, when m=i or is divisible by two or more different prime numbers. We have n/(<o=m, where d runs through all divisors of m, from which it follows by Theorem 8, B, that the quotient — — = /(m) is different from I only when m is a power of a prime number, in which case it is equal to this prime. For a derivation by another method of the other properties of the <f> functions from the single one that %+(d) = \nt\ t see Kronecker, Vorlesungen uber Zahlentheorie, Vol. I, pp. 245, 246. Also for another independent proof that +(ofr)»+(a)+(*), if a be prime to b, see the same, p. 125. § 6. ^-Functions of Higher Order. 1 The theory of the <£- function may be generalized as follows : By <f> n (m) we denote the number of sets of n integers of a com- plete residue system, mod m>, such that the greatest common divisor of the integers of each set is prime to m, two sets being different if the order of the integers in them be different. For example, let w = 6; then 1,2,3,4,5,6 1) 1 Cahen: pp. 36, Z7- Bachman : Niedere Zahlentheorie, pp. 91, 93. THE RATIONAL REALM — CONGRUENCES. 55 will be a complete residue system, mod 6. All possible sets of two numbers each that can be formed from the numbers i) are I, I h 2 h 3 I, 4 i, 5 i,6 2, I 2, 2 2, 3 2, 4 2, 5 2, 6 3, i 3, 2 3, 3 3, 4 3, 5 3,6 4, i 4, 2 4, 3 4, 4 4, 5 4,6 5, i 5, 2 5, 3 5, 4 5, 5 5,6 6, i 6,2 6, 3 6,4 6, 5 6, 6 Of these there are twelve sets the greatest common divisor of the numbers of each of which is not prime to 6 ; they are 2, 2 ; 2, 4 ; 2, 6 ; 3, 3 ; 3, 6 ; 4, 2 ; 4, 4 ; 4, 6 ; 6, 2 ; 6, 3 ; 6, 4 ; 6, 6. There are therefore twenty- four sets, the greatest common divisor of the numbers of each of which is prime to 6. Hence 4> 2 (6)=24. It can be shown that «^-l*K^Kf)('-Rp)-"( i T.Kp) ( where p lf p 2 , '",p r are the different prime factors of m. The following theorems can also be proved : i. If m^p, a prime number, then <f>n(P) = \p\ n —^ ii. // \m\ > 2, <f> n (m) is even. iii. If m x and m 2 be two integers prime to each other, then 4> n (m x m 2 ) = <£n( w i) < M w *)- iv. If d run through all divisors of m, $<j> n (d) = \m\ n . Ex. Let m = 6, and n = 2 ; then 2 (6)=6 2 (i-J 2 )(i-J 2 )=2 4 . $& THE RATIONAL REALM CONGRUENCES. § 7. Residue Systems Formed by Multiplying the Numbers of a Given System by an Integer Prime to the Modulus. Theorem 9. // m ly m 2f --,m m be a complete residue system, mod m, and a be prime to m, then am^ am 2 , •••, am m is also a com- plete residue system, mod m. The integers am x , am 2 , • • • , am m are incongruent each to each, mod m, for from ami = amj, modm, it would follow that, since a is prime to m, miz==mj, mod w, which is contrary to the hypothesis that m x ,m 2 ,--,m m form a complete residue system, mod m. The integers am x , • • •, am m are, moreover, | m'\ in number. They form, therefore, a complete residue system, mod m. Cor. If r 1 ,r 2 ,---,r4 >(m) form a reduced residue system, mod m, and a be prime to m, then ar t$ •••,ar^ m) is also a reduced resi- due system, mod m; for ar lf •••, ar^m) are incongruent each to each, mod m, prime to m and cf>(m) in number. Ex. Since — 9, 2, —17, 14, 15, —4, —13, 8, 19, 20 constitute a complete residue system, mod 10, and 3 is prime to 10, — 27, 6, —51, 42, 45, —12, —39, 24, 57, 60 is also a complete residue system, mod 10. Likewise since — 9, —17, —13, 19 is a reduced residue system, mod 10. — 27, —51, —39, 57 is also a reduced residue system, mod 10. If p be any prime number and a any integer prime to p, it is evident from the above that there exists an integer a x such that aa L = 1, mod p. We call a x the reciprocal of a, mod p. THE RATIONAL REALM CONGRUENCES. 57 § 8. Fermat's Theorem as Generalized by Euler. Theorem io. If m be any rational integer and a any rational integer prime to m, then a0 (w) ss i } mod m. Let fit ?2> ' ' '} r <b(nt) i) be a reduced residue system, mod m. Then since afit Of* • • •, ar Um) 2) is also a reduced residue system, mod m, each integer of 2) is congruent to some integer of 1), mod m, that is, we have ar 1 , modw, 3) where rf 1 >rj 2 >'",rj <f> , m) are the integers 1), though perhaps in a different order. Since t$ v t$^ '"> r u im ar e the integers 1), we have Multiplying the congruences 3) together, we have a0 (m) P = P, modw, 4) where P is prime to m, since each of its factors is prime to m. Hence, dividing both members of 4) by P, we have o0 (w) ==i, modw. 5) If m== ± p n , where p is a prime, we have a i\P\-D\ P \^^ lf m od/ 1 , 6) and, in particular, when m = p a bl-i = Ij mod p. 7) ' If p be positive, 7) becomes aP- 1 ^!, mod/>; 8) that is, if p be a positive prime number, and a an integer not divis- ible by p } aP' 1 — 1 is divisible by p. This is the form in which the theorem was enunciated by Fermat. 1 1 This theorem was published by Fermat in 1670, without proof. Euler was the first to give a proof. He gave two : Comm. Acad. Petrop. VIII, 1741, and Comm. Nov. Acad. Petrop. VII, p. 74, 1761. 58 THE RATIONAL REALM CONGRUENCES. Ex. i. Let m = i5; a = 2; then <f>(is) =S. From 5) it follows that 2 <f>(m) = 2* = 1, mod 15 ; that is, 256 = 1, mod 15. Ex. 2. Let P = 7; a = — 3- From 7) it follows that (— 3) 8 =i, mod 7; that is, 729 = 1, mod 7. Ex. 3. Let m=zpn = f; a = 2; then <f> (3*) =2-3 = 6. From 6) it follows that 2 6 be 1, mod 9 ; that is, 64== 1, mod 9. On account of the great importance of Fermat's theorem, we shall give for the form 8) a second proof, depending upon the binomial theorem. If aP^==a, mod/>, 9) where p is a positive prime, hold for every integral value of a, then aP' 1 ^ 1, modp holds when a is prime to p. We shall show now that 9) holds for all integral values of a. We see that 9) holds when a=i. If, therefore, we can show that a sufficient condition that 9) shall hold for a = a 1 -\-i is that it shall hold for a = o x , 9) will hold for all positive integral values of a. We have by the binomial theorem (a + 1 )p = op + pav~- ^titzzl}^ + . . . + '<*-'>"*% + 1. V ' ' ' t ' 12 "l-2 '•• (/> i) ' From § 10 we know that all coefficients in this expansion are integers. Hence since p occurs as a factor in the numerator of the coefficient of every term except the first and last, and, since the denominators of these terms contain only factors that are prime to p, the coefficient of every term except the first and last is divisible by p, and we have (a4-i)P = aP+i, modp, for every integral value of a. THE RATIONAL REALM CONGRUENCES. 59 Therefore ((% -f I ) p = of + i, mod p, whence assuming that 9) holds for a=.a 1 ; that is, 0/ = ^, mo&p, we have (a ± + i)p^a x + 1, mod/) ; that is, 9) holds for a = a 1 -{- 1, if it holds for a = a t . But 9) holds for 0=1. Hence 9) holds for every positive integral value of a. Moreover, since every negative integer is congruent to some positive integer, mod p, 9) holds also for all negative integral values of a. Fermat's theorem in the form 8) is an immediate consequence of the theorem that we have just proved. § 9. Congruences of Condition. Preliminary Discussion. The congruences which we have so far considered may be com- pared to arithmetical equalities, the values of the quantities in- volved being given and the congruence simply expressing the fact that the difference of the two numbers is divisible by the modulus. We shall now consider congruences which hold only when special values are given to certain of the quantities involved ; that is, the values of these " unknown " quantities are determined by the condition imposed by the congruence; for example, let x be determined by the condition that its square is to be congruent to 2, mod 7. We have x 2 =z 2, mod 7, and see easily that we must have xmz or —3, mod 7/ To develop the theory of congruences of condition, it is neces- sary to introduce the conception of the congruence of two poly- nomials with respect to a given modulus; thus, if f{x lf x 2 , •••,*») be a polynomial 1 in the undetermined quantities x 1 ,x 2 , --,x n with rational integral coefficients, we say that f(x lf x 2 , •••,#„) is iden- tically congruent to with respect to the modulus m, if all its coefficients be divisible by m. 1 We shall understand by a polynomial in n undetermined quantities Xi,x 2 , ••-,xn a rational integral function of Xi t x 2 , ••-,xn whose coefficients, unless the contrary be expressly stated, are rational integers. 60 THE RATIONAL REALM CONGRUENCES. This relation is expressed symbolically by f(x 1 ,x a ,---,x H )=o, modw. 1 Two polynomials f(x ly x 2 , •••, x n ) and 4>(x lf x 2f '-,x m ) are said to be identically congruent to each other, mod m, if their differ- ence be identically congruent to o, mod m, or what is the same thing if the coefficients of corresponding terms in the two poly- nomials be congruent; that is, in symbols f(x 1 ,x 2 ,'~,x H )==<f>(x 1 ,x 2 ,~-,x n ), mod™, if f(x 1 ,x 2 ,-.,x n )—<l>(x 1 ,x 2 ,--,x n )=o, modw. For example, we have 8x 2 — 2xy -f- 6y + I = 2x 2 + xy — 2, mod 3, since 6x 2 — 3*3' -f-6y — 3=0, mod 3, or, in other words, since 8 = 2, — 2=1, 6=0, and 1= — 2, mod 3. If f(x lt x 2 , •••', x n ) ^<f>(x lf x 2y '-',Xn), mod m, and a lf a 2 , ~,a n be any n integers, then evidently f(a 1 ,a 2 ,'--,a n )===<l>(a 1 ,a 2 ,--,an). modiw. If, however, all the coefficients of f(x lf x 2 , •■-,x n ) be not congru- ent, mod m, to the corresponding coefficients of <f>(x lf x 2t -",x n ), we do not have in general f(a lt a 2f ••-,a w )=^(a 1 ,a 2 , • -,o»), mod in, 1) for ever>- set of integers o lf a 2 , • • •, a*. The demand that x x , x 2 , ■ • •, .r„ shall have such values and only such that 1) will hold is expressed by writing f{x Xi x 2i ~-,Xn)=4>(x li x 2 ,---yXn), mod ;;/. 2) Any set of integers satisfying 1) is called a solution of 2). The determination of all such sets, or the proof that none exist, is called solving the congruence 2). It is customary to say, how- ever, that a congruence is solvable or unsolvable according as it has or has not solutions. We call 2) a congruence of condition. 1 The symbol == is read " is identically congruent to." THE RATIONAL REALM CONGRUENCES. 6 I If a lf a 2 ,-",an and b r ,b 2 ,---,b n be two sets of n rational in- tegers and a 2 ?=b 2 k mod m, 3) then by § 1, v, ■ f(a lf a 2 ,-", a„) = f(b lt b 2 , — ,V) a mod M, and <^(a 1? a 2 , •••,o„)=<^(& 1 , & 2 , ■••,&,), modw. Hence, if a x , a 2 , • • •, a„ be a solution of 2), b lf b 2l •••,&» is also a solution. Two solutions so related are, however, looked upon as identical. In order that two solutions may be counted as different, it is necessary and sufficient that there shall be in the one solution a value of at least one unknown which is incongruent, mod tn, to the value of the same unknown in the other solution ; that is, the n relations 3) must not hold simultaneously. It is evident from the above that in order to solve any con- gruence, as 2), it is sufficient to substitute for the unknowns the \tn\ n sets of values obtained by putting for each unknown the \tn\ numbers of a complete residue system, mod m, and observe which values of f(x lf x 2 , -",x n ) so obtained are congruent to the corre- sponding values of <f>(x lf x 2J •••,.r„), mod m. There being only a finite number, |w|", of possible solutions, we can by this process always completely solve any given congruence. If the congruence have the form f(x x ,x 2 , •••,jt„)=o, mod mi, and a lf a 2 , "- 9 Om be a solution, then f{x x ,x 2 , ••• i x n ) is said to be zero, mod m, for these values of x x ,x 2 , '-' f x n . Ex. Let us consider the congruence f(x,y) =2x* — xy + y — 2f + 1=0, mod 3. 1 4; 1 In order to avoid confusion, we shall use throughout this book the symbol = instead of as to denote algebraic identity. 62 THE RATIONAL REALM CONGRUENCES. Putting for x and y, the numbers — i, o, I of a complete residue system, mod 3, we obtain nine values of / (x, y). /(0,-l)=-2, /(l,_l)=I, /(_!,_!)=_!, /(o, o) = i, /(i, o) =3, /(— i, o) =3, /(o, i) =o, /(i, i) si, /(-i, i) =3, Four of these values /(o, i), /(i,o), /( — i,o), and /( — i, i) are con- gruent to o, mod 3. Hence the solutions of 4) are: .r= 1, y==o x==—i, y==o t mod 3. By the degree of a polynomial, mod m, we shall understand the degree of the term, or terms, of highest degree, whose coefficient, or coefficients, are not divisible by m. A reduced polynomial, mod p, is one whose coefficients are all numbers of the residue system, 0,1, •••,/> — 1. § 10. Equivalent Congruences. Addition and Multiplication Transformations. Two congruences A Oi, *%* ' "4 ** ) '■■ f»(f 1* *%, ••'*, **) j mod m, 1 ) and <t> 1 (x 1 ,x 2 ,---,x n )=<f> 2 (x 1 ,x 2 ,'-',x n ),modm, 2) are said to be equivalent when every solution of the first is a solu- tion of the second, and every solution of the second is a solution of the first. In solving a congruence, as in the case of algebraic equations, we proceed under the assumption that a solution exists and look upon the congruence as an identity in the values of x 1} x 2 , --,x n that satisfy it, though as yet unknown. Looking then upon 1) as an identity in these unknown values of x x ,x 2 , --,x n , we con- sider what operations can be performed upon 1 ) that will produce another identity 2) such that each of these identities is a neces- sary consequence of the other. Operations of which this is true we shall call reversible operations. Referring to §1, we see that there are two such operations: first, if 1 ) be the given congruence and F x (x 1 ,x 2 ,--,x n )=F 2 (x 1 ,x 2 ,'-' i x n ), modw, 3) THE RATIONAL REALM CONGRUENCES. 63 be any identical congruence, mod m, in x ly x 2 , -",x n , we can add 3) member by member to i), obtaining + F s (x 1 ,x t ,— f x n ), modw, a congruence equivalent to i). By means of this transformation, we can transpose any term with its sign changed from one member of a congruence to the other, and can thus reduce any congruence, as i), to an equiva- lent congruence of the form f(x lf x 2 , '■•,x„)==o, modm, 4) whose second member is o. We shall hereafter assume the con- gruences with which we deal to have been reduced to this form. We may also by this transformation reduce the coefficients of f(x lf x 2 , '-',Xn) to their smallest possible absolute values, mod m, and thus lessen the labor of solving the congruence. Ex. The congruence 14X 4 — wx 3 -f- zx 2 + 7 X — 12 ==0, mod 7, 5) is equivalent to the congruence — 3** -f- 2X 2 + 2 == o, mod 7, which has two roots x a — 1 or 2, mod 7, and these are therefore the roots of 5). A second operation which, when performed upon any congru- ence, as 1) or 4), yields an equivalent congruence, is the multipli- cation of both members of the congruence by any integer, a, prime to the modulus ; that is, the congruences f(x x , x 2 ,---,x n )==B o, mod m, and &f( x v x 2> '"> x n) =0, mod m, where a is prime to m, are equivalent. Conversely, we may divide all the coefficients of a congruence by any integer prime to the modulus, obtaining an equivalent congruence. Ex. The congruences I5^y — 2ixy -f 3y 2 + 9 == 0, mod 35 64 THE RATIONAL REALM CONGRUENCES. and S^y — 7xy + y 2 + 3 = 0, mod 35 are equivalent. As a special case of the multiplication transformation, as we shall call the second of the above transformations, we have the multiplication of the congruence f(x 1 ,x 2 ,---,x n )=o, modm, by — 1 ; that is, the change of sign of each of its coefficients. § 11. Systems of Congruences. 1 Equivalent Systems. So far we have considered only single congruences ; that is, the unknown quantities are subjected to a single condition. We can, however, as in the case of algebraic equations, subject them to two or more conditions simultaneously; that is, x lt x ±i "*,x n may be required to satisfy simultaneously the congruences fi(x 1 ,x 2J ---,x n )^o, modWi, ft Oi, *2> * ' '> *n) = o, mod m 2 , fr(*is *»••'• *j *n) s= o, mod m r . By a solution of such a system of congruences we understand a set of values of x\,x 2 ,'-',x n which satisfy simultaneously all the congruences. Two solutions, a ly a 2 , ~- y a n and b lt b 2 , •••,&«, are considered dif- ferent when and only when the nr congruences a 2 = b 2 ^,mod m& mod m 2 , ••-, mod m, a n =b n ~ are not satisfied simultaneously. Two systems of congruences are said to be equivalent when each solution of the first system is a solution of the second and each solution of the second is a solution of the first. It is evident that any one of the congruences of the system can be transformed 1 See Stieltjes: Essai sur la theorie des Nombres. THE RATIONAL REALM CONGRUENCES. 65 into an equivalent congruence by the transformations of the last article and the system so obtained will be equivalent to the origi- nal system. If the moduli be the same, we can obtain an equiva- lent system by adding two congruences and taking the new con- gruence together with the r — 2 of the original ones not used and either one of those used. Thus the system /iC*if*i» •••»•*•) ■ B *i modw, ) x is equivalent to the system fx C*u *»'•"> x * ) = °> m od m, f 1 (x 1 ,x 2 , ■■-,x n ) +f % {x*>x 9 ,—,*%) =0, modw, or, more generally, if a x ,a 2 be any two integers prime to m, 1) is equivalent to the system f x (x x ,x 2 , •••,*„) ==0, modw, oJi(x lt x 2t ■■•,x n ) -\-a 2 f 2 (x lf x 2 , •••,#„) =0, modw. Ex. Let the given system be 4-^ — 3? + 7* = 5 ] $x + y — 3z==2 L mod 17. 2) x — 43/ — s==i J Multiplying the third congruence first by — 4 and then by — 5, and adding it to the first and second respectively, we obtain the system I3y + ii2= il 2iy-j- 2z = — 3 Lmodi7, 3) ' * — 43? — 2 == 1 J that is equivalent to 2). Adding the first and second congruences of 3), we obtain the equiva- lent system 132 = — 2I 2iy-f 22 = — 3 L mod 17. x — 4y— z= 1 J The congruence 132^ — 2, mod 17, has the single solution z == — 8, mod 17, that substituted in 2iy-|-2.s== — 3, mod 17, gives yas — 1, mod 17. Substituting these values of y and 8 in # — 4y — z ;= 1, mod 17, 5 66 THE RATIONAL REALM CONGRUENCES. we have x^6, mod 17. We obtain therefore as a solution of the given system x == 6, y == — 1, 8 ess — 8, mod 17, a result easily verified by substitution in the original system. The method of solution shows that this is the only solution (see §13). § 12. Congruences in One Unknown. Comparison with Equations. The general congruence in one unknown has the form /( x) — a x n + & x x n - x + • • • + a n = o, mod m. 1 ) If r be a rational integer such that f(r) =0, modm, r is called a root of 1). The degree of 1) is, as has been said, the degree of the term of highest degree whose coefficient is not divisible by m. Such a congruence presents many analogies to the equation a x n + a x x n ^ + ••• + <z n = o; 2) for example, to the addition to both members of the equation of the same function of the unknown corresponds the addition to the members of the congruence of any functions of the unknown which are identically congruent with respect to the modulus, and to the multiplication of the equation by any quantity not a func- tion of the unknown corresponds the multiplication of the con- gruence by any integer prime to the modulus. If m be a prime number the congruence presents still other striking analogies with algebraic equations, these analogies being absent in the case of a composite modulus. For example, consider the two congruences of the second degree O— i)0 — 3) so, mod 7, 3) and (x — i)(.r — 3) =0, mod 12. 4) We see that 3) has two roots, 1 and 3, while 4) has four roots, 1, 3, 7 and 9; that is, 3) has a number of roots equal to its degree, while 4) has more roots than its degree. The analogy with algebraic equations in the case of the prime THE RATIONAL REALM CONGRUENCES. 67 modulus is as evident as is the lack of analogy in the case of the composite modulus. We shall see later that no congruence of the form 1) with prime modulus can have more roots than its degree. The reason for this difference in the case of the above example is seen to be that, if a be any integer, the product (a — i) (a — 3) is divisible by a prime number, as 7, when and only when one of its factors is divisible by this prime, a statement no longer true when the modulus is composite ; that is, a product is zero, mod m, when and only when one of its factors is zero, mod m, if m be a prime number, but not otherwise. We shall, therefore, in the discussion of the general congruence of the form 1 ) confine our- selves first to the case in which the modulus is a prime and shall then show that the solution of any congruence of the form 1) with composite modulus can be reduced to the solution of a series of congruences of the same form with prime moduli. Although striking analogies between congruences and algebraic equations have already been pointed out, while others will be observed later, it is important to note an essential difference between them. In the case of an algebraic equation it is the same thing to say that all the coefficients of an equation are zero or that it is satisfied by every value of the unknown quantity, each of these properties implying the other. In the case of congruences, however, although, if the coefficients be all congruent to zero with respect to the modulus, the con- gruence is, of course, satisfied by any integral value of the unknown, on the other hand, it is not true in general that, if a congruence be satisfied by all integral values of the unknown, that all of its coefficients are divisible* by the modulus. For example, as is easily seen from Fermat's theorem, the congruence x p — .r==o, modp, where p is a prime, is satisfied by every integral value of x; but its coefficients are not all divisible by p. The reason for the dif- ference will be shown later. We shall see also that, although a 68 THE RATIONAL REALM CONGRUENCES. congruence of the form i ) with prime modulus can not have more roots than its degree, it can have less; for example, the three congruences x 3 — 2x 2 — x -\- 2 == o, mod 5, x 3 + 2 * 2 — 2x -\- i =3 o, mod 5, x 3 + 4X 2 + x -f- 1 33= o, mod 5, that are all of the third degree and have the same prime modulus, 5, have respectively three roots, 1, — 1, and 2, one root, — 2, and no root. Before taking up the general congruence in one unknown, we shall consider that of the first degree. § 13. Congruences of the First Degree in One Unknown. The most general congruence of the first degree can be written in the form ax^=b, mod m. We shall consider first the case where a is prime to m. Theorem ii. The congruence \ ax==b, modra, where a is prime to m, has always one and but one root. If we put for x successively the \m\ integers m lf m 2 , '••,m m of a complete residue system, mod m, we obtain \m\ integers am lf am 2 , ~-,am m , that also -constitute a complete residue system (Th. 9), and it is evident that one and but one of these integers, say ami, will be congruent to b, mod m. Hence the congruence has always one and but one root, Wj. We can evidently solve any congru- ence of this form by this method. Ex. Let the given congruence be 3*3= — 5> mod 14. 1) Taking as a complete residue system, mod 14, the integers o, 1, 2, 3, •••, 13, and putting x equal to these values in succession, we have $x = 6, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39. The only one of these integers that is congruent to — 5, mod 14, is 9 ; that is, 3-3 33= — 5, mod 14. Hence ^==3, mod 14, is the single root of 1) THE RATIONAL REALM CONGRUENCES. 69 By means of Fermat's theorem we can find a general expres- sion for the root of a congruence of the above form. Since a is prime to m, we have a 0(m) == I} modm, which multiplied by b gives ba<t> (m} ^b, modm, or aba ( P (m) - 1 ^b, modm. Hence £?a0 (m)_1 is the root of the congruence ax^b, modm, where a is prime to m. This is one of the few cases in the theory of numbers where the quantity sought can be expressed as an explicit function of the given quantities. Ex. The root of 3*== — 5, mod 14, is jt== — 5-3* (14)_1 , mod 14; that is, *=== — 5-3 5 == — n==3, mod 14. We shall now consider the general case where a is any integer that may or may not be prime to m. Theorem 12. The necessary and sufficient condition for the solvability of the congruence ax^E=b, modm, is that b shall be divisible by the greatest common divisor, d, of a and m, and when this condition is fulfilled, the congruence has exactly \d\ incongruent roots. Let a = a x d and m = m x d, where a x is prime to m x . From ax==b, modm, 2) we have a x dx = b + km x d. Hence b must be divisible by d; that is, b = b x d is a necessary condition that 2) can be solved. This gives a x dx = b x d -\- km x d, 3) or a x x = b x , modm^ 4) yO THE RATIONAL REALM CONGRUENCES. Since a x is prime to m x , 4) has a root (Th. 11). Moreover, all roots of 4) are also roots of 2) ; for from 4) follows 3) and hence 2). Therefore the divisibility of b by d is a sufficient as well as necessary condition for the solvability of 2). We see also that not only are all roots of 4) roots of 2), but all roots of 2) satisfy 4) and are therefore integers of the form r + km x , where r is a root of 4). We ask now how many of these roots are in- congruent to each other, mod m; that is, how many incongruent roots has 2) ? Any two roots, r + k ± m v r + k 2 m x , of 4) are con- gruent, mod m, when and only when r -\- k 1 m x — (r -f- ^wj — nm, where n is an integer ; that is, if ( k x — k 2 ) m 1 — nm x d, or k x — k 2 = nd, or k x ^k 2 , modd. Hence, in order that the roots of 2) shall be incongruent, it is necessary and sufficient that the values of k shall be incongruent, mod d. If we put, therefore, for k the \d\ integers of a complete residue system, mod d, for example, o, 1,2, • • •, \d\ — 1, we shall obtain all the incongruent roots of 2), namely r, r + m t , r + 2w x , •••,r+(|d| — 1) m x . They are evidently \d\ in number. Ex. Consider the congruence 1 2.x ^ — 20, mod 56. 5 ) Here d — 4. Dividing by 4 we have 3* = — 5, mod 14, a congruence whose root has already been found to be — 11. Therefore the roots of 5) have the form — 11 -{- 14&, and are four in number. They are — 11, 3, 17 and 31. § 14. Determination of an integer that has certain residues with respect to a given series of moduli. Let us consider first the case in which the required integer has to satisfy two such conditions ; that is, we are to determine x so that we have simultaneously THE RATIONAL REALM CONGRUENCES. 7 I x = a x , modwu i) and x = a 2 , modm 2 . 2) All integers satisfying 1) have the form x = a x -\-m x y, where y is an integer. Since x must also satisfy 2), y must satisfy the condition m x y = a 2 — a x , mod w,. 3) By Th. 12 for 3) to have a solution, it is necessary and sufficient that a 2 — a x shall be divisible by the greatest common divisor, d, of m x and m 2 . If this requirement be fulfilled and y be one root of 3), every root, y, of 3) must satisfy the condition that IS, y=y Q J r -jy v where y x is any integer. All integers satisfying both 1) and 2) have therefore the form mm x =a x + m x y + —^y x \ . . ., m,m„ that is, x = a x + m x y , mod * . Hence if x be any integer satisfying both 1) and 2), all and only those integers satisfy both 1) and 2) that are congruent to x with respect to the least common multiple of the moduli of 1) and 2). By an easy extension of this method we obtain the common solution, if any exist, of the n congruences x==a x , modwj, x==a 2 , mod w 2 , 4) x^=a n , mod ///„, and we see that, if x be an integer satisfying all these congru- ences and / the least common multiple of the moduli, .r == x , mod /, gives all the common solutions of the system 4). The general 72 THE RATIONAL REALM— CONGRUENCES. problem of determining whether any given system of congruences of the form ax^=b, mod m, have common solutions and of find- ing them, if they exist, can be solved by the above method. When the coefficients of x are prime to the moduli the congruences can evidently be reduced to the form x=c, mod m, and we have the case just treated. If the moduli be prime each to each, / = m 1 m 2 •" m n and the congruences 4) always have a common solution. We shall now give another solution of this problem for the special case last mentioned. This solution is interesting on ac- count both of its symmetry and some important deductions that can be made from its form. W 7 e have then to determine the common solutions of the congruences 4) , the moduli m lf m 2 , •-, m n being prime each to each. We determine first for each modulus, mi, an^auxiliary integer, bi, such that bi is congruent to 1 with respect to the modulus nti and is divisible by each of the other moduli, and hence by their product ; that is, we determine b lf b 2 ,--,b n so that b x =i, modm^ and ^ = 0, mod m 2 m 3 •••#»«, & 2 ==i, mod w 2 , and b 2 = o, mod m 1 m s • • • m n , b n =i, modm n , and b n =o, mod m 1 m 2 • • • m n _^. It is evident that this can always be done, for we have in the case of b x from the second condition b 1 = tn 2 m 3 -- m n c lf and it only remains to determine a value for c t in accordance with the condition m 2 m z - - - m n c 1 mm 1, mod m lf that is always possible since m 2 m s • • • tjt n is prime to m x . Having found these auxiliary integers, we put r = a 1 b 1 + a 2 b 2 -f ••• + a n b n , THE RATIONAL REALM CONGRUENCES. 73 and shall show that the common solutions of 4) are the integers satisfying the congruence jrasr, mod m x m 2 • • • m„. 5) If x satisfy 5), then jir = r, mod mi, 6) and, since all the auxiliary integers except bi are divisible by mi, from 6) it follows that x = ciibi, mod Mi, and hence, since bi^i, mod m^, we see that .r==ai, mod m^ Hence every integer, that satifies 5), satisfies each of the con- gruences 4). Moreover, every integer, that satisfies each of the congruences 4), satisfies 5), for, if x be such an integer, then from x ==cii, mod nii, and r = di, modwj, we see that x — r = o, modWij that is, x — r is divisible by each one of the moduli m lf m 2 , • • •, tn n , and hence, since they are prime to each other, by their product. Therefore x ==r, mod m 1 m 2 ••• m n . Hence the integers satis- fying 5) are all the common solutions of 4). It will be observed that the auxiliary integers b x ,b 2 ,-",b n are entirely independent of a 1} a 2 , ■••,a n , being dependent only on the moduli. Ex. It is required to find the common solutions of the congruences x==2, mod 11, x===4, mod 15, x==9, mod 14. To calculate the auxiliary integers bi, b 2 , b 3 , we have &x = 2ioci==i, mod 11, fr 2 =i54c 2 =i, mod 15, b 3 = 165^3^ 1, mod 14, and hence Ci==i, modn, ^ = 210, c 2 ^ 4, mod 15, £ 2 = 6i6, c z ^g, modn, & 3 =i485. Therefore r = 420 + 2464 + 13365 = 16249, 74 THE RATIONAL REALM CONGRUENCES. whence ;r== 16249, 11^2310^ or x^yg, mod 2310, a result that is easily verified. We observe now two important facts concerning r, that are direct consequences of the symmetrical method of its formation. First, if for a lt a 2 ,---,a n be put the integers of complete residue systems with respect to the moduli m 1 ,m 2 , •••,*»», respectively, the resulting values of r form a complete residue system, mod /, for we obtain thus |/| values of r and they are incongruent each to each, mod /. To show this, let two values of r be r' as a 1 f b 1 + a % % + * • • + CLn'bn, and r" = a x "b x + a 2 "b 2 + ■ ■ • + *•%* where we do not have simultaneously a x ' ss a/', mod m lf a 2 ' = a 2 ", mod m 2 , ■••, a*'ss a n ", mod m n ; that is, in order that the two values of r be different we must have at least one of the a"s, such as a/, in r' incongruent, mod mi, to the corresponding a", a/', in r". Let at sfs a" , mod m%. If r' = r", mod/, it would follow that r' ^ r", mod Wi, and hence also ai'bi^=ai"b{, mod w», or, since 5i=i,modw{, ai'^ai", mod Wi, that is contrary to our supposition. The two values of r are therefore incongruent with respect to the modulus /. In the second place, if we select from the system of values of r just formed those which are formed by putting for a lf a 2 ,---,a„, the integers of reduced residue systems with respect to the moduli m lf m v -••,tn n respectively, the resulting values of r form a reduced residue system, mod /. We have already shown that these values of r are incongruent each to each, mod /. It re- mains to be shown that all and only those values of r that are prime to / occur in the system as formed. If one of these values THE RATIONAL REALM CONGRUENCES. 75 of r, as r', =a 1 f b 1 + • • • + a n 'b n , have a prime factor, p, in com- mon with I, then some one of the moduli, as mi, must have this factor in common with r, and since r' = ai, mod Mi, ai and nti would have the common factor p, which is contrary to the hypothesis that «#' is an integer of a reduced residue system, mod mi. Hence all values of r obtained above are prime to /. More- over, when a value of r, as r', is prime to /, a/, a 2 , '-,an are each prime to their respective moduli, for, if any a, as a,-', have a factor p in common with its modulus, then since r' = a/, mod Wi, r' would have the factor p in common with m\ % and hence with /. Hence all values of r, that are prime to /, occur in the above sys- tem, and it is therefore a reduced residue system, mod /. Ex. Let mi =6, m 2 = 5, we have bx = 5C1 & i, mod 6, and b 2 = 6c 2 ^=i, mods, whence Cis=5, mod 6, and c 2 ^ i, mod 5. Then b x = 25, and b 2 = 6, whence r = 2501 -)- 6a 2 . Putting for ai the values 1, 5 and for a 2 the values 1, 2, 3, 4, that is, the integers of reduced residue systems, mod 6, mod 5, respectively, we have for the resulting values of r 31, 37, 43, 49, 131, 137, 143, 149, that, being all prime to 30 and in number 0(30), = 8, constitute a reduced resi- due system, mod 30. This method of forming a reduced residue system shows us at once that the number of integers in such a system, mod m 1 m 2 ■ • ■ m n , where m 19 m 2 , ••-, m n are prime each to each, is equal to the prod- uct of the numbers of the integers in the reduced residue systems for each of the moduli tn lt m 2 , • • •, m n . We obtain therefore a new proof of Th. 4 ; that is, that 4>(m x m 2 ■■■ m n )=cf,(m 1 )<l>(m 2 ) ■■■<f>(m»), where m x ,m 2 , '",m n are prime each to each. y6 THE RATIONAL REALM CONGRUENCES. We shall proceed to the discussion of the general congruence of the wth degree in one unknown with prime modulus and shall first develop briefly the theory of the divisibility of polynomials with respect to a prime modulus. § 15. Divisibility of one Polynomial by another with respect to a Prime Modulus. Common Divisors. Common Multiples. If p be any rational prime number we have the following definition: A polynomial, f(x), is said to be divisible with respect to the modulus p by a polynomial <f>(x) when there exists a poly- nomial Q(x) such that f(x)mQ(x)*(x), modp. We say that <f>(x) and Q(x) are divisors or factors, mod p, of f(x), and that f(x) is a multiple, mod p, of <f>(x) and Q(x). We also say that /(•*") is resolved, mod p, into the factors <f>(x) and Q(x). The degree of a polynomial, mod p, is the degree of the term of highest degree whose coefficient is not divisible by p. The sum of the degrees of the factors of f(x) is evidently equal to the degree of f(x). Ex. It is easily seen that x* + 3x A — 4* 3 + 2= (2X 2 — 3)(3x* — * 2 + 1), mod 5. Hence 23? — 3 and 3x s — x 2 -f- 1 are divisors, mod 5, of x 5 + 3** — 42* + 2. We have as direct consequences of the definition of divisibility : i. // f ± (x) be a multiple, mod p, of / 2 (^') and f 2 (x) be a mul- tiple, mod p, of f 3 (x), then f±{x) is a multiple, mod p, of f s (x) f or more generally, if each polynomial of the series f 1 (x), f 2 (x), '">fn(x) be a multiple, mod p, of the one immediately following, then each polynomial of the series is a multiple, mod p, of all that follow. ii. // f t (x) and f 2 (x) be multiples, modsp, of f{x), then f\{ x ) +/2(- r ) an d fi_(x) — f 2 ( x ) are multiples, mod p, of f(x), or more generally, if f x (x) and f 2 {x) be multiples, mod p, of f(x), and F 1 (x),F 2 (x) be any two polynomials, then F 1 (x)f 1 (x) -f- F 2 (x)f 2 (x) is a multiple of f(x). If two or more polynomials f 1 (x),f 2 (x),---,f n (x) be divis- ible, mod p, by a polynomial <f>(x), <f>(x) is said to be a common THE RATIONAL REALM CONGRUENCES. JJ divisor, mod p, of f x (x),f 2 (x), •••,/„(». If a polynomial f(x) be a multiple, mod p, of two or more polynomials <£i(-r),<k>(- l ')> •■■,<f>„(x), f(x) is said to be a common multiple, mod p, of § 1 6. Unit and Associated Polynomials with Respect to a Prime Modulus. Primary Polynomials. We ask now whether there exist polynomials that with respect to a modulus p divide all polynomials. Evidently those have this property that are of degree o and are ^ o, mod p ; that is, the ra- tional integers not divisible by p, for they are divisors, mod p, of I and i divides every polynomial. Furthermore, these are the only polynomials having this property, for no polynomial, f(x), of degree higher than the oth can divide, mod p, all polynomials, for it can not divide i, since then the sum of the degrees of the divisor and the quotient, mod p, would be greater than o, the degree of i. We call the rational integers, excluding those divisible by p, the unit polynomials, mod p, or briefly, units, mod p, and since two polynomials that are congruent, mod p, are considered as identical, we can take as the units, mod p, the integers of any reduced res- idue system, mod p, for example, I, 2, •••, \p\ — i. Thus the unit polynomials, mod 7, are 1, 2, 3, 4, 5, 6. Two polynomials which differ only by a unit factor, mod p, are called associated polynomials and are looked upon as identical in all questions of divisibility, mod p. If two polynomials, f 1 (x), / 2 (■*"), are eacn associated, mod p, * with a third polynomial, they are associated with each other; for if f 1 (x)=af 3 (x), modp, 1) and f 2 (x)=bf 3 (x), modp, 2) where a and b are units, mod p, then, multiplying 2) by b lf the reciprocal, mod p, of b, we have btft00mf 9 (x), modp, and hence from 1) f 1 (x)=ab 1 f 2 (x), modp, where ab 1 is a unit, mod p. yS THE RATIONAL REALM CONGRUENCES. Two polynomials, that are associated, mod p, are evidently of the same degree and each is a divisor, mod p, of the other. Conversely, if two polynomials be each divisible, mod p, by the other, they are associated. Two polynomials that have no common divisor, mod p, other than the units are said to be prime to each other, mod p. Any polynomial, f(x), has \p\ — I associates, mod p. Of these one and only one has as the coefficient of its term of highest degree i. This one is called the primary associate, mod p, of f(x). For example, the six polynomials x 3 -\-2x — -3, 2x 3 + 4X — 6, 3^r 3 + 6x — 2, 4x 3 + x — 5, 5^ 3 + 3.r— 1, 6x 3 + 5x — 4, are associated, mod 7, and x 3 + 2x — 3 is the primary one. §17. Prime Polynomials with respect to a Prime Modulus. Determination of the Prime Polynomials, mod p, of any Given Degree. A polynomial that is not a unit, mod p, and that has no divisors, mod p, other than its associates and the units, is called a prime polynomial, mod p. If it has divisors, mod p, other than these it is said to be com- posite, mod p. To find the primary prime polynomials, mod 3, of any given degree we may proceed as follows, considering all polynomials to be reduced. All polynomials of the first degree are evidently prime. Hence primary prime polynomials of the first degree, mod 3, are three in number, namely x, x -\- 1, x -f-2. The reduced primary polynomials, mod 3, of the second degree are nine in number, namely X 2 , X 2 + X, X 2 + 2X, X 2 +I, X 2 + X-\-l, X 2 -\-2X+I, X 2 + 2, X 2 + X + 2, X 2 -^-2X-\-2. From the three primary polynomials of the first degree, we can form the six composite polynomials of the second degree THE RATIONAL REALM CONGRUENCES. 79 .i- 2 = X 2 , x (x + I ) = x- + X, (x + i) 2 ==x 2 + 2x + i, ".r(^ + 2)=^r 2 + 2jr, mod 3. These being the primary composite, polynomials, mod 3, of the second degree, we see that X 2 +I, X 2 + X + 2, X 2 + 2X + 2, are the primary prime polynomials, mod 3, of the second degree. In like manner we see that there are nineteen composite poly- nomials of the third degree, mod 3, and hence eight prime poly- nomials of the third degree, mod 3, since there are in all twenty- seven reduced primary polynomials of the third degree, mod 3. It can be shown that, when n is greater than 1, the number of prime polynomials, mod p, of the nth. degree is y n n n - (p n — 2/^ -f 2/^2 — 2/^3 ^ ) f where q x ,q 2 i<lz, "> are the different prime factors of n. This expression being always different from o, it follows that there exist prime polynomials, mod p, of any given degree. 1 § 18. Division of one Polynomial by Another with Respect to a Prime Modulus. Theorem 13. // f(x) be any polynomial and <f>(x) be any polynomial not identically congruent to o, mod p, there exists a polynomial Q(x), such that the polynomial f(x) — Q(x)4>{x)==R{x), modp, 1) is of lower degree than <f>(x). The operation of determining the polynomials Q(x) and R(x) is called dividing f(x) by <f>(x), mod p. We call Q(x) the quo- tient, and R(x) the remainder in the division, mod p, of f(x) by <f>(x). We shall prove the existence of Q(x) andi?(^r) by giving a method for their determination. 1 H. J. S. Smith : p. 153. Borel et Drach : pp. 49, 50. Bachmann : Niedere Zahlentheorie, pp. 372, 373. 80 THE RATIONAL REALM CONGRUENCES. Let /(*) = a x n + a,*"- 1 + • • • + o n , 4> (x) = b x m + hi**-* -\ h b m be any two polynomials and let & =j=o, mod p. We shall consider first the case in which b is I, and shall then show that the general case can be reduced to this one. Since b is i, we can divide f(x) by <f>(x) as in ordinary division until we get a remainder R(x) of lower degree than <f>(x), the quotient being Q(x). We have then from which follows at once i). We can now reduce to this particular case the general case in which b has any value not divisible by p. Let c be the recip- rocal, mod p, of b ; then c </>0) ==</>i 0')> rnodp, 2) where <f> x (x) is a polynomial the coefficient of whose term of highest degree is 1 when reduced, mod p. Dividing f(x) by <f) 1 (x) as above, we have f{x)^Q{x)4> 1 {x)+R{x), modp, and hence, making use of 2), f(x)mc Q(x)^(x)+R(x), modp, where c Q(x) andi^(^r) are the quotient and remainder required. 1 The above theorem plays the same role in the theory of the divisibility of polynomials with respect to a prime modulus that Th. A does in that of rational integers. Ex. Let it be required to divide, mod 7, f{x) = 5 X S — 2X t -{-2X S — 5X 1i + 2X+I, by <f>(x)=3x B + x*—$x —2. 1 See also Cahen : p. 70, Borel et Drach : p. 33, and Bachmann : Niedere Zahlentheorie, p. 368, concerning the division of one polynomial by another with respect to a prime modulus.. THE RATIONAL REALM CONGRUENCES. 8 I Since 5 is the reciprocal, mod 7, of 3, we have 0i O) =50 O) =.r 8 + 5* 2 + 3* — 3, mod 7. 3) Dividing /O) by 0iO) as in ordinary algebraic division, we have Sx 5 — 2jt 4 + 2X 3 — sx z + 2x J ri — (sx 2 — 27X + 122) (V + 5** + 3* — 3) = —519^ — 445^ + 367, whence, reducing coefficients, mod 7, 5jt 5 — 2* 4 + 2X 3 — 5^- 2 + 2X + 1 — (— 2X 2 + * + 3) (** + 5^ + 3* — 3) ^ — * 2 + 3^+3, mod 7, or, making use of 3), 5* 3 — 2x* + 2X 3 — sx 2 + 2x + 1 — 5 (— 2X 2 4- * + 3) (3-** + x 2 — sx — 2) = — «" + 3* + 3, mod 7 ; that is, j^— a**4-2jr*-f- 5«* + ar+ i— (— 3X 2 — 2x + 1) (3^ + ^ — 5* — 2) = — x 2 + 3x + 3, mod 7, where — 3X 2 — 2X + 1 and — .r 2 -f 3* + 3 are the required polynomials Q(x) and R(x). § 19. Congruence of two Polynomials with Respect to a Double Modulus. Two polynomials, f±{x), f 2 (x). are said to be identically con- gruent to each other with respect to the double modulus p, <f>(x), where p is a prime number and <f>(x) a polynomial, if their differ- ence, f x {x) — f 2 {x), is divisible, mod p, by <f>(x); that is, in symbols /*(*)■»/,(*), modd/>, 4>(x), 1) if A(*)— /,(*) £§(?(*)*(*), modp, 2) or, in other words, if AC*)— U(x)=*Q{x)*{x)+F{xy.p, 3) where Q(x) and F(x) are polynomials. It should be observed that 1), 2) and 3) all express exactly the same relation between the polynomials f 1 (x), f 2 {x) and <f>(x) and the prime number p, but, just as in the case of congruences between integers, 1) places this relation before us in a more illuminating manner than does either 2) or 3). 6 82 THE RATIONAL REALM CONGRUENCES. The fact that f(x) is divisible, mod p, by <f>(x) is expressed in the above notation by writing /(*)=o, modd/>, <j>(x). Ex. From § 15, Ex., we have * 8 + 3* 4 — 4-^ + 2 = 0, modd5, 2X 2 — 3. We have as consequences of the above definition just as in the case of integers, the double modulus p, <f>(x) being understood, throughout. h(i)mf,(s) f*(*)mf t {*)> f t (x) ■/,(*). f l (x)mf t (x) U(s)mf£x) momial, F(x)f 1 (x)^F(x)f 2 (x). f x {x)mf t (x) F 1 (x)=F 2 (x), Ux)-F^x)^f t {x)-F t {x), f t (x)mf t (x), The results corresponding to v, • • •, ix, § 1, follow easily. § 20. Unique Factorization Theorem for Polynomials with respect to a Prime Modulus. We shall now show that a polynomial can be resolved in one and but one way with respect to a prime modulus, p, into prime factors, considering always associated factors as the same. The proof will be closely analogous to that of the corresponding l If and then ii. if and then iii. // and F(x) be any then iv. // and then and, in particular, if then THE RATIONAL REALM CONGRUENCES. 83 theorem for rational integers. We begin by stating the following theorem which is an immediate consequence of the definition of divisibility. Theorem 14. If f{x) = Q(x)<f>(jy) -f R(x), mod p, every polynomial that divides, mod p, both f(x) and <f>(x) divides both <f>(x) and R(x), and vice versa; that is, the common divisors, mod p, of f(x) and <f>(x) are identical with the common divisors, mod p, of <f>(x) and R(x). By means of this theorem and Th. 13 we can now prove the theorem which is the basis of the unique factorization theorem. Theorem 15. // f 1 (x), f 2 (x) be any two polynomials and p a rational prime, there exists a common divisor, D(x), mod p, of . fi(*)> fi( x ) sucn that D{x) is divisible, mod p, by every common ' divisor, mod p, of f 1 (x) } f 2 (x), and there exist two polynomials ' 0i(- r )> <f>2( x )> sucn that We may evidently assume f 2 (x) of degree not higher than f x (x). Dividing f ± {x) by f 2 (x), mod p, we can find two polynomials Q x (x), f 3 (x), such that f 1 (x) = Q 1 (x)f 2 (x)+f s (x), modp, f & (x) being of lower degree than f 2 {x). Dividing f 2 (x) by f s (x), mod p, we have f 2 {x)=Q 2 (x)f z (x)+f A (x), modp, where f 4 (x) is of lower degree than f 3 (x), and similarly h{x) = Q z {x)f,{x)+U(x), modp, /n_ 2 (*) ==Q n _ 2 (>)/«-! (*) +/«0), ™od p, fn. 1 {x)^Q n . 1 {x)f n (x), modp, a chain of identical congruences in which we must after a finite number of steps reach one in which the remainder, f n+1 (x), is o, mod p, since the degrees of the remainders continually decrease. 84 THE RATIONAL REALM CONGRUENCES. By Th. 14 the common divisors, mod p, of f n (x) and fn- x (x) are identical with those of ftut(x) and /„_ 2 (x), those of f n - x (x), fn- 2 (x) with those of fn-t(x), fn. s (x), and finally those of / 3 (.r), f 2 (x) with those of f a (x), f x (x). But fn(x) is a common divisor, mod p, of f n (x) and f n -i(x) and is evidently divisible by every common divisor of f n (x) and f«-i(x). Hence f n (x) is the desired common divisor D(x), mod />, of f x (x) and/ 2 (». If now we substitute the value of f s (x) in terms of f ± (x), f 2 (x) obtained from the first of these congruences in the second and the values of / 3 (.r) and f 4 (x) in terms of' f x (x) , f 2 (x) in the third and continue until the congruence / n _ 2 (*) =Q„_ 2 (*)/«-i (-0 +/nO), modp, is reached, we shall obtain the congruence f x (x)4> t (x) +f t (x)4>t(x)mD(x) s modp. Cor. If f x (x), f 2 (x) be two polynomials prime to each other, mod p, there exist two polynomials <f> x (x), <f> 2 (x) such that AOO^O) + / 2 (.r)* 2 (*) = i, modp. In this case D(x) is an integer a not divisible by p, and we have two polynomials ^(x), $ 2 (- r ) such that f t (x)^(x) +f t (x)* 2 (x)w&a, modp, whence, multiplying by the reciprocal of a, mod p, we obtain /i(-0<£iO) + /*(*)**(*) ■■ h modp. It will be noticed that this corollary corresponds to Th. B, while Th. 15 corresponds to the corollary to Th. B, the order of proof here being reversed. The corollary could have been proved first as before. 1 Theorem 16. // the product of two polynomials, f x {x) , f 2 (x) , be divisible, mod p, by a prime polynomial, P(x), at least one of the polynomials, f x {x), f 2 {x), is divisible, mod p, by P(x). Let f 1 (x)f 2 (x) = Q(x)P(x),modp, 1) 1 Laurent : Theorie des Nombres Ordinaires et Algebriques, p. 120. THE RATIONAL REALM CONGRUENCES. 85 where Q(x) is a polynomial, and assume f x (x) not divisible, mod p, by P{x). Then f x {x) and P(x) are prime, mod p, to each other and by the last theorem there exist two polynomials, <\> 1 {x) 1 <f> 2 (x), such that £(*)*>(*) +P(*)^(*) « I, mod p. 2) Multiplying 2) by f 2 (x), we have &(*)/,(*)*{*) +/ 2 (-r)P(.r)^(.r)=/ 2 (^), mod/», and therefore, making use of 1), P(.r)(Q(.r)^(.r) + / 2 (jt)^(jt) ) ==f 2 (x), modp, where Q(x)<f> 1 (x) -j-f 2 (x)<f> 2 (x) i s a polynomial. Hence / 2 (-r) is divisible, mod p, by P(x). Expressed in the double modulus notation this theorem is: If fi( x )> /*(«*) be any two polynomials and P{x) a prime poly- nomial, mod p, and if h(x)f 2 (x)=o, modd/>, P(x), then either f 1 (x)^o, modd/>, P(x), or /,(.r)=o, modd/>, P(x). Cor. 1. If the product of any number of polynomials be divis- ible, mod p, by a prime polynomial P(x), then at least one of the polynomials is divisible, mod p, by P(x). Cor. 2. If neither of two polynomials be divisible, mod p, by a prime polynomial P(x) % their product is not divisible, mod p, by P(x). Theorem 17. A polynomial, f(x), can be resolved in one and but one way into a product of prime polynomials, mod p. Let f(x) be any polynomial. We shall take f(x) in its reduced form, mod p, for the sake of convenience, this assumption in no wise limiting the generality of the proof. Let the degree, mod p, of f(x) be n. If f(x) be prime, mod p, the theorem is evident. If f(x) be not prime, it has a divisor, <f>(x), mod p, and we have f(x)=<j>{x)*{x), modp, 86 THE RATIONAL REALM CONGRUENCES. where <f>(x), &(x) are polynomials neither of which is a unit and the sum of whose degrees is n. If <f>(x) be not a prime polynomial, mod p, then , +(jc)Wk^i*y* x {x) % modp, where ^(x), &i(x) are polynomials that are not units and that have degrees whose sum is equal to the degree of <f>(x). If <f> 2 (x) be not a prime polynomial, mod p, we proceed in the same manner and, since the degrees of the factors form a decreas- ing series of positive rational integers, we must after a finite number of such factorizations reach in the series 4>(x), <t>i(x), <f> 2 (x), • • • a prime polynomial P x (x), mod p. We have then f(x)=P 1 (x)f 1 (x), modp. Proceeding similarly with f x (x) in case it be not prime, mod p, we obtain U{x)wiP*(j*)f % {s)t modp, where P 2 (x) is prime, mod p, and hence f(x)=P 1 (x)P 2 (x)f 2 (x), modp. Continuing this process, we must after a finite number of such factorizations reach in the series f(x), f t (x), f 2 (x),-- a prime polynomial P n (x), mod p. We have then f(x)=P 1 (x)P 2 (x) --'Pn(x), modp, where P 1 (x),P 2 (x), •-■,P n (x) are all prime, mod p; that is, f(x) can be resolved, mod p, into a finite number of prime factors. It remains to be shown that this resolution is unique. Suppose that f{symQi(s)Q 9 (s) "-Qm(x), modp, be a second resolution of f(x) into prime factors, mod p. Then P 1 (x)P 2 (x) -P n (x)=Q 1 (x)Q 2 (x) -Qm(x), modp, 3) and it follows from Th. 16, Cor. 1 that at least one of the Q(jtr)'s, say Q x (x), is divisible, mod p, by P 1 (x) and hence is associated, mod p, with P x (x) ; that is, Q 1 (x)^a 1 P 1 (x), modp, where a x is a unit, mod p. THE RATIONAL REALM — CONGRUENCES. 87 Dividing 3) by P x (x), mod p, we have P 2 (x) -'-P n (x)^ ai Q 2 (x) "-Q n (x), modp. 4) From 4) it follows that at least one of the remaining Q(x)'s must be associated, mod p, with P 2 {x). Dividing 4) by P 2 (x), mod p, and proceeding as before, we see that with each P{x) there is associated, mod p, at least one Q (x) and, if two or more P(x)'s are associated, mod p, with one another, at least as many Q(x)'s are associated, mod p, with these P(x)'s and hence with one another. In exactly the same manner, we can prove that with each Q(x) there is associated, mod p, at least one P{x) and, if two or more Q(x)'s are associated, mod p, with one another, at least as many P(x)'s are associated, mod p, with these Q(x)'s and hence with one another. Hence, considering two associated factors as the same, the resolutions are identical; that is, if in the one resolution there occur e factors associated, mod p, with a certain prime polynomial, there will be in the other resolution exactly e factors associated, mod p, with the same prime polynomial. We can now evidently write any polynomial, f(x), in the form /(#)«(*»<*))*'(/»,(*))•... (P.Cjt))-, mod ^ where P 1 (x),P 2 (x), •••,P„(or) are the unassociated prime fac- tors, mod p, of f(x). If we take P 1 (x),P 2 (x), ••',P n {x) primary, the resolution is absolutely unique. The representations of the greatest common divisor and least common multiple given for rational integers are easily extended to polynomials. § 21. Resolution of a Polynomial into its Prime Factors with respect to a Prime Modulus. The resolution of a polynomial, f(x), into its prime factors, mod p, may be effected by dividing, mod p, f(x) by each of the prime polynomials of the first degree x,x — i,---,x — p -\- i,(p being taken positive) in turn until either a polynomial is found that divides f(x), or it is determined that f(x) is divisible by none of them. 88 THE RATIONAL REALM CONGRUENCES. Suppose that f(x) is divisible, mod p, by x — a x and that the quotient is f x (x). We proceed in the same way with f x (x) until we have found all the prime, mod p, factors of the first degree oif(x). Suppose that f(x) = (x — a 1 )(x — a 2 ) ••• (x — a„)f 2 (x), modp, where f 2 (x) has no factor, mod p, of degree lower than the second. The prime factors, mod p, of the second degree of f 2 (x) can next be determined in the same manner, then those of the third degree, etc. In case, however, we do not know the prime, mod p, polynomials of the second degree, we can simply determine whether f 2 (x) is divisible, mod p, by any polynomial of the second degree. If it is, such a polynomial is evidently a prime, mod p, polynomial, for f 2 (x) contains no factors, mod p, of degree lower than the second. The same method can be applied to the deter- mination of the prime factors of higher degree. § 12. The General Congruence of the nth Degree in one Unknown and with Prime Modulus. Theorem 18. // r be a root of the congruence f(x) — a x n -f OjX*- 1 + ■•• -\-a n = o, mod p, i ) f(x) is divisible, mod p, by x — r, and conversely, if f{x) be divis- ible, mod p, by x — r, r is a root of i). Dividing, mod p, f(x) by x — r, we have f(x)m{x-~ r)*(*)+*(r), mod/>, whence, since r is a root of i), R(r) =o, modp, and hence f(x) = (x — r)<f>(x), modp; that is, f(x) is divisible, mod p, by x — r. The converse is evident. If f(x) be prime, mod p, the congruence i) evidently has no roots. The converse is, however, not true; that is, f(x) may be 1 Borel et Drach : p. 36. THE RATIONAL REALM CONGRUENCES. 89 composite, mod p, but i) have no roots, for the prime, mod p, factors of f(x) may all be of higher degree than the first. This theorem gives us another method for determining the factors, mod p, of the first degree of any polynomial in x. Some of these factors may be alike and we are led therefore to say that r is a multiple root of order e of i), if f{x) be divisible, mod p, by (x — r) e , but not by (x — r) e+1 . If therefore r lf r 2i '",r m be the incongruent roots of i) of orders e t ,e 2 ,---,e m respectively, we have f(x) = (x — r x )*(x~ r 2 ) e *... (x — r m ) e '-f 1 (x), modp, where f 1 (x) is a polynomial having no linear factor, mod p, and whose degree, s, is such that ^1 + ^2 + "• +em + s=n, where n is the degree of f(x). Counting a multiple root of order e of i) as e roots, we see that 1) has exactly as many roots as f(x) has linear factors, mod p, and obtain the following important theorem: Theorem 19. The number of roots of the congruence f(x) = a x n + a x X** *\ + 0»seO, mod p, where p is a prime number, is not greater than its degree. Cor. 1. If the number of incongruent roots of a congruence with prime modulus be greater than its degree the congruence is an identical one. Cor. 2. If the congruence f(x) =0, mod p, 2) have exactly as many roots as its degree and cf>(x) be a divisor, mod p, of f(x), then the congruence (f>(x) =0, mod p, has exactly as many roots as its degree; for f(x)==<l>(x)Q(x), modp, where Q(x) is a polynomial in x, and every root of the congruence 2) is a root of either the congruence <£(»=== o, modp, 3) gO THE RATIONAL REALM — CONGRUENCES. or of the congruence Q(jr)==o, mod p. 4) Moreover, the sum of the degrees of 3) and 4) is equal to the degree of 2). If, therefore, <f>(x) had fewer roots than its degree, then Q (x) must have more roots than its degree, which is impos- sible. Hence the corollary. § 23. The Congruence x0 (m) — 1 = 0, mod nu Although in the case of congruences of degree higher than the first the theorem just given tells all that we know in general regarding the number of their roots, still there is one important case in which the number of roots is always exactly equal to the degree of the congruence. Theorem 20. The congruence x <p(m-> — i===o, modw, 1) has exactly as many roots as its degree. The <]>(nt) integers of a reduced residue system, mod m, evi- dently satisfy 1). Moreover, since by §1, ix, two integers con- gruent, mod m, have with m the same greatest common divisor, and the greatest common divisor of 1 and m is 1, every root of 1) must have with m the greatest common divisor 1, that is, be prime to m. Hence the number of roots of 1) is exactly equal to <£(m), its degree. Ex. The congruence #* C10) — 1 ==0, mod 10, or x* — 1^0, mod 10, has the four roots 1, 3, 7, and 9. Cor. If d be a positive divisor of p — 1, the congruence x d — 1 = o, mod p, where p is a prime, has exactly d roots; for x d — / is a divisor of x*- x — 1 and hence by Th. 19, Cor. 2, we have the corollary. Since the congruence x p — x===o, modp, THE RATIONAL REALM — CONGRUENCES. 9 1 has the p roots o, i, 2, • • -, p — 1 equal in number to its degree, we have the identical congruence x p — x = x(x — %)(x — 2) ••• (x — p — 1), mod p. Ex. X 1 — x = x(x — i)0 — 2)0 — 3)0 — 4)0 — 5)0 — 6), mod 7. § 24. Wilson's Theorem. The result just obtained gives us a proof of the following inter- esting theorem. Theorem 21. If p be a prime number and r x ,r 2 , •••,r$ (p) be a reduced residue system, mod p, then Vi • • • Uip) + 1=0, mod p. By the previous section we have evidently x<t>^ — i = {x — r 1 ){x — r 2 ) ••• (* — !>(„,), modp, from which, putting x = o, we have — i = (— r 1 )(— r 2 ) ••• (— r+ m } t modp, whence, since <f>(p) is even except when p = 2, Vj "-^(p)) + 1 = 0, modp, which evidently holds also when p = 2. 1 Ex. Let p = 5, and take as a reduced residue system, mod 5, the integers — 2, — 1, 1, 2. Then (— 2)(— i).i.2 + i = 5 = o, mod 5. This theorem is a particular case of the following more general theorem that is due to Gauss. 2 If r i> r 2>'"> r <i>(m) be a reduced residue system, mod m, the product r x r 2 -~ r^ (m) is congruent to — i, mod m, when m = 4, p n or 2p n , where p is an odd prime, and is congruent to 1, mod m, when m has any other value. The two following examples will illustrate this theorem; for its proof see references given above. Ex. 1. Let m = f, and take as a reduced residue system, mod3 2 , — 4, — 2, — 1, 1, 2, 4; then (— 4)(— 2)(— 1). 1-2-4 = — 64 = — 1, mod3 2 . 1 See Matthews, § 16, for another proof of this theorem. 2 Gauss: Disq. Arith., Art. 78. Dirichlet-Dedekind : §38. Bachmann: Niedere Zahlentheorie, p. 170. Cahen : p. 103. 92 THE RATIONAL REALM CONGRUENCES. Ex. 2. Let m = 15, and take as a reduced residue system, mod 15, — 7, — 4,-2,-1, 1, 2, 4, 7; then (—7) (—4) (—2) (— 1) -1-2.4.7 = 3136= 1, mod 15. As a special case of Th. 21 we have the following: If p be a positive prime number and the product of all positive integers less than p be increased by 1, the result is divisible by p; that is, (p — 1 ) ! + 1 sb o, mod p. The theorem was first stated in this form by Waring in his " Medi- tationes Algebraicae " (1770) and ascribed to its author, Sir John Wilson. The converse of the original form is true ; that is, // the product of all positive integers less than a given integer, m, be increased by 1 and the result be divisible by m, then m is a prime number. This is easily seen to be true; for, if m = ab, where neither a nor b is a unit, then (m — 1) ! is divisible by a, whence we have (m — 1 ) ! + I W^ o, mod m. For example 5 ! + 1 = 121 ^o, mod 6. Wilson's theorem gives therefore an unfailing method for deter- mining whether any given integer is a prime number. It is, how- ever, obviously of no practical use on account of the immense labor of the numerical reckoning when m is large. § 25. Common Roots of Two Congruences. The common roots of two congruences f 1 (x)=o, modp, and / 2 (jr)=o, modp, are evidently the roots of the congruence <f>(x) =0, modp, where 4>( x ) ls tne greatest common divisor, mod p, of f x {x) and / 2 (.r). Since the congruence x p — ;r = o, modp, 1) has for its roots the numbers of a complete residue system, mod p, the incongruent roots of any congruence /(,r) =0, modp, THE RATIONAL REALM CONGRUENCES. 93 will be the roots of the congruence <f>(x) =0, modp, 2) where cf>(x) is the greatest common divisor, mod p, of x? — x and f(x). This gives us another method of determining all the incongruent roots of any given congruence with prime modulus. The congruence 2) will always have as many roots as its degree, since the congruence 1) has as many roots as its degree and <j>(x) is a divisor, mod p, of x? — x. Ex. To find the roots of the congruence x l — sx 3 — x 2 -\-2x — 6 = 0, mod 7, 3) by the above method, since o is not a root of the congruence, we need only find the greatest common divisor, mod 7, of x l — 3x* — x 2 -\-2x — 6 and x e — 1. This -greatest common divisor is x 2 — 3X -\- 2, and the congruence x 2 — 3^ + 2^0, mod 7, has the roots 1 and 2, that are therefore the incongruent roots of 3). § 26. Determination of the Multiple Roots of a Congruence with Prime Modulus. The multiple roots of the congruence f(x) =0, modp, 1) may be determined by a method exactly analogous to that em- ployed for determining the multiple roots of an algebraic equation. Thus let P(x) be a prime function, mod p, and let f(x) be divis- ible, mod p, by (P(x)) e but not by (P(x)) e+1 ; then f(x)==(P{x)YQ{x), modp, or, what is the same thing, f{x) = {P(x)YQ{x)+pF(x), 2) where F(x) and Q(x) are polynomials in x and Q(x) is prime, mod p, to P{x). Differentiating 2), we have f(x) = (P(x)y->(eP'(x)Q(x) + P(x)Q'{x)) + pF'(x), where .P'(.r), Q'(x') and F'(x) are polynomials in x. Hence f(x) = (P(x))^Q 1 (x), modp, 94 THE RATIONAL REALM CONGRUENCES. where Q x (x) is a polynomial in x and is moreover not divisible, mod p, by P(x), for Q x (x)=eP'(x)Q(x) +P(x)Q'(x), where P'{x) is of lower degree than P(x) and Q(x) is prime, mod p, to P(x). Therefore f(x) is divisible, mod p, by the prime factor P(x) exactly once less than f(x) is divisible by P(x). In particular, if f(x) be divisible, mod p, by {x — r) e , but not by (x — r) e+1 , then f(x) is divisible, modp, by (# — r) e_1 but not by (x — r) e . Hence the theorem: Theorem 22. // the congruence f(x) = 0, modp, have a multiple root r of order e, the congruence f(x) = 0, modp, has the multiple root r of order e — 1. If the greatest common divisor, mod p, of f(x) and f(x) be <f>(x), then the roots of the congruence 4>0)=o, modp, 3) if it have any, will be the multiple roots of 1) and each root of 3) will occur once oftener as a root of 1) than as a root of 3). It may happen, of course, that f(x) and f(x) have a common divisor, <f>(x), mod p, and yet 1) has no multiple roots. In this case the repeated prime factors, mod p, of f(x) are of higher degree than the first, and <f>(x) therefore contains no factor of the first degree, mod p. Ex. Let the given congruence be f(x)=2x 8 — * -|- 1 ^ °> mod 5- 4) We have f(x)=6x 2 — i^^r — 1, mod 5, and the greatest common divisor, mods, of /(■*") and f(x) is x -\- 1. The congruence x + 1 ^ o, mod 5, has the root — I. Hence the congruence 4) has two roots — 1. Dividing f(x) by (x + i) 2 , we have f{x) =2(x -\- i) 2 (x — 2), mods, and see that f(x) has the third root 2. THE RATIONAL REALM — CONGRUENCES. 95 § 2j. Congruences in One Unknown and with Composite Modulus. The solution of a congruence of the form f( x) = a x n + a x x n ~ x + " + o»930, mod m, i ) where m = m x m 2 • • ■ m t , m lf m 2 , "• ntt being integers prime each to each, can be reduced to the solution of the system of t congruences, f(x) s=o, modMj,! f(x) =o, mod w 2 , ; : r 2) f(x)z==o, modwf.J Every root of i) is evidently a root of each of the congruences 2), and conversely any integer, that is simultaneously a root of each of the congruences 2), is a root of«i). If therefore a lf a 2 , --,at be roots of the congruences 2) and r be chosen so that r = a 19 mod m lt ' r==a 2 , modm 2 , I : : l 3) rz=a t , mod m t , - then r is a root of 1). Since m lf m 2 , • • •, mt are prime each to each, it is, by §14, always possible to find r so as to satisfy the conditions 3). Let b lt b 2 ,---,bt be auxiliary integers selected as in § 14; then r = a 1 b 1 + a 2 b 2 + • • • + a t b t , mod m 4) is a root of 1), and, if the congruences 2) have respectively hyh>'"Jt incongruent roots, then by §14 1) has l x l 2 --l t incon- gruent roots, that are obtained by putting in 4) for a r ,a 2i --,at respectively the l^,l 2 ,-",h roots of the congruences 2). In particular, if any one of the congruences 2) have no root, then 1) has no root. Ex. The solution of the congruence ** + 3* 3 + 3** + 3* + 2 = 0, mod 30, 5) 96 'THE RATIONAL REALM CONGRUENCES. can be reduced to the solution of the two congruences x* + 3X 3 -f sx- -+- $x -\- 2 == o, mod 6, 6) and x* + 3x 3 + sx 2 + 3x + 2 = o, mod 5. 7) The roots of 6) are — 2, — 1, 1, 2 and those of 7) are — 2, —2, — 1, 2. The roots of 5) are then r.Oj = — 2, — I, I, a. 1 r== 25* + 6a 2 , mod 30. j a __ 2 __^ 2 that gives as the roots of 5), — 13, — II, — 8, — 7, — 2, — 1,2,4,7,8, 13, 14. If now we suppose m to be resolved into a product of powers of its different prime factors, that is, m=p 1 ei p 2 e2 ••• p r er , where p lt p 2 , "',pr are different primes, then the solution of 1) is reduced to the solution of n congruences of the form /(.r)=o, modp e . 8) We shall now show tfiat the solution of 8) can be made to depend upon the solution of the congruence /0)=o, mod^" 1 , 9) where the modulus is a power of p one degree lower than that of the modulus of 8), and thus be made to depend eventually upon the solution of the congruence f(x) =0, modp, whose modulus is a prime. Let x 9 be a root of 9) ; then all integers of the form x + p e ~ 1 y, where y is an integer, are roots of 9). Furthermore, since all roots of 8) are roots of 9), if 8) have roots they must be of this form. Putting in 8) x = x + p e ~ x y, 10) we have f(x -j- p e ~ x y) ^ o, mod p e , or, expanding f(x + p^y), / /K) +f(^)P e - 1 y+ f -^P 2e -'Y- + -^f mod J-. 11) Since f(* ) =0, mod/> e_1 , 1 See Example § 14. THE RATIONAL REALM — CONGRUENCES. 97 we have f(*o)=cp e ~ 1 , and hence, dividing each term of n) by /> e_1 , c + fWj+^f¥ + "^o, mod?, whence we have c -\-f(x )y = o, modp, 12) as a necessary and sufficient condition that y must satisfy in order that the root, x + p e ~ x y, of 9) may also be a root of 8). There are three cases to be considered: i. If f(x 9 )qk6 s modp, there is always one and but one value, y , of y that satisfies 12) and this gives one value only of x + p e ' x y^ tnat satisfies 8). ii. If f(x )==o, modp, and c^£o, modp, there is no value of y satisfying 12) and hence no value of x of the form x + p e ' 1 y satisfying 8) ; that is, 8) has no root. iii. If f(x )^=o, modp, and c = o, modp, then 12) is an identical congruence and consequently 12) has \p\ solutions, mod p, from which by substitution in 10) we obtain \p\ solutions of 8). 1 Ex. The roots of the congruence x* — 8x* + 9** + 9* +14 = ©, mod 5 2 , 13) if any exist, must satisfy the congruence x* — 8-r 3 + 9x~ + 9* + 14 aa 0, -mod 5, whose roots are 1 and 2, and hence be of the form i+5y or 2-j-Sy. Substituting 1 + 5y and 2 -f- 5y in 13), we obtain respectively 5+ 7y = o, mod 5, 14) and 4 — io:y = o, mods. 15) From 14) we have y as o, mod 5, and from 15) ;y==i, mods, that give 1 and 7 as the roots of 13). 1 See Cahen :• pp. 96-103. 7 98 THE RATIONAL REALM CONGRUENCES. §28. Residues of Powers. // a be prime to m, and b^a f f mod m, where t is a positive integer, b is said to be a power residue of a with respect to the. modulus m. For example, since 4==3 2 , mod 5, we say that 4 is a power resi- due of 3 with respect to the modulus 5. Two power residues of a which are congruent to each other, and hence to the same power of a, mod m, are looked upon as the" same. A system of integers such that every power residue of a, mod m, is congruent to one and only one integer of the system, mod m, is called a complete system of power residues of a with respect to the modulus m. Ex. Every power of 5 is congruent, mod 6, to 1 or 5. Hence 1, 5 constitute a complete system of power residues of 5, with respect to the modulus 6. These integers may evidently be selected from among the in- tegers of any reduced residue system, mod m. For convenience they are usually taken from the system 1,2, •••, \m\ and we may indeed define a complete system of power residues of a, mod m, as being the smallest positive residues that the successive powers of a, a°= 1, a 1 , a 2 , a 3 , •••,#', •■• give when divided by m. The more general definition given above will, however, serve our purposes better as it will admit of direct extension to realms of higher degree than the first, while the latter does not. We shall now investigate certain questions relating to power residues, and, in particular, the important one as to when a com- plete system of power residues of an integer a, mod m, is also a reduced residue system, mod m. The following table gives the power residues of all numbers of a reduced residue system, mod 13, with respect to this modulus. In order to calculate the residue of a k , it is not necessary to raise a to the &th power, but only to multiply the residue of o fc-1 by a and then take the residue of the product with respect to m. m== 13. THE RATIONAL REALM CONGRUENCES. 99 I 2 3 4 5 6 7 S 9 io ii 12 I 4 9 3 12 IO IO 12 3 9 4 i i 8 i 12 s 8 5 5 i 12 5 12 I 6 9 io 5 2 II 8 3 4 7 12 I 12 I I 12 12 12 12 I I 12 I I I I 3 4 S 7 6 5 9 io 2 12 I 5 i 12 5 5 S s I 12 I IO 3 9 12 4 4 12 9 3 io i i 7 9 io 8 ii 2 5 3 4 6 12 We ask now, what is the smallest value t a of £ other than o for which we have a'= i, modw. That t a always exists and is ^<j>(m) is evident from Fermat's theorem, that gives, since a is prime to m, a 0(m)== i } mod w. Giving t a the above meaning, we say that the integer a appertains to the exponent t a with respect to the modulus m. We see from the table that 2, 6, 7, 1 1 appertain to the exponent 12; that is, ^(13). " 4, 10 appertain to the exponent 6 5, 8 appertain to the exponent 4 r, mod 13. 3, 9 appertain to the exponent 3 12 appertains to the exponent 2 It is evident that, if a= b, mod m, then a and b appertain to the same exponent, mod m. Theorem 2^. If the integer a appertain to the exponent t a , mod m, then the t a powers of a, 1, a, a 2 , --^V 1 , are incongruent each to each, mod m. Let a s and a s+r be any two of the powers 1). If a 8+r ==a 8 , modm, then, since a is prime to m, a r z= 1, modw. 1) 3) 100 THE RATIONAL REALM CONGRUENCES. But r < t a and hence 3) is impossible, since a appertains to t a . Therefore 2) is impossible. Theorem 24. // a appertain to the exponent t a , mod m, any two powers of a with positive exponents are congruent or incon- gruent to each other, mod m, according as their exponents are congruent or incongruent, mod t a . Let a 81 , a s * be any two powers of a, s lf s 2 being positive integers, and let s± s=a q x t a + r 1} s 2 = q 2 t a + r 2 , where q lf q 2 are positive integers and o^r x <t a , o^r 2 <t a , r^r 2 . 4) If a&**^»a** i+rt , modwz, 5) then a r i==a r 2, mod m, 6) whence, since a is prime to m, a r 1 -r 2 ^ Ij rnodwt. But from 4) we have o^r 1 — r 2 <t a , and hence, since a appertains to t a , mod m, r ± = r 2 . 7) Therefore' s ± = s 2 , modf a , 8) is a necessary condition for a s i = a S2 , mod m. 9) Moreover, from 8) follow in turn 7), 6) and 5). Hence 8) is also a sufficient condition for the existence of 9). We have therefore a 1 as a ta+1 hbs a 2ta+1 == a 3ta+1 h= • a 2 = a ta+2 sb a 2ta+2 = a 3 * a+2 ss • a'"" 1 = a 2ta ~ x be a 3 *"" 1 se a 4 * " 1 = , modw. This is known as the law of the periodicity of the power resi- dues. It can be verified by an examination of the table, p. 99, 5 1 = 5' 5' 5 2 as 5 C 5 10 5 3 = 5 7 S 11 = 4 = 8=12 y , mod 13, *=£_ 9 L mod 4 . THE RATIONAL REALM CONGRUENCES. IOI where we see, for example, that 5 appertains to the exponent 4, mod 13, and we have 2 = 6 = IO 3 = 7=11 J Theorem 25. The exponent, t a , to which an integer a apper- tains with respect to the modulus m, is always a divisor of <f>{m)} Since a0< w > == 1 = a , mod m, we have by Th. 24, <f>(m) = 0, modt a . Theorem 26. If two integers, a lf a 2 , appertain, mod m, to two exponents, t lt t 2 , that are prime to each other, then their product, a x a 2 , appertains, mod m, to the exponent, tj 2 . Let a x a 2 appertain, mod m, to an exponent t, then (a 1 a 2 y*smi s modm. 10) Raising both members of 10) to the t x power, we have a^ a ^xt == lf m0( | m But a ± tlt aa 1, mod m, and hence a,,* 1 '— 1 * modm, and therefore, since a 2 appertains to the exponent t 2 , mod m, t x t must be a multiple of t 2 (Th. 24). Whence, since t x and t 2 are prime to each other, it follows that t is a multiple of t 2 . In like manner we can show that t is a multiple of t x . Therefore t, being a multiple of t t and t 2 , that are prime to each other, is a multiple of their product t x t 2 . Hence the smallest pos- sible value of t for which 1) will hold is tj 2 , and a±a 2 appertains to this exponent, mod m. Ex. We see from the table, p. 99, that 12 and 3 appertain, mod 13, to the exponents 2 and 3 respectively, and that their product 36(^ 10, mod 13) appertains to the exponent 6. Limiting ourselves now to the case in which the modulus is a 1 For a proof of this theorem not dependent upon Fermat's theorem, see Mathews, p. 18. 102 THE RATIONAL REALM CONGRUENCES. prime number, p, we ask whether there are integers appertaining to every positive divisor of <f>(p) and, if so, how many. Before proving the theorem, that will answer this question in its entirety, let us examine the table, p. 99, and see how matters stand when /> — 13. The positive divisors of ^(13), = 12, are 1, 2, 3, 4, 6 and 12. To 1 appertains the single integer 1, To 2 appertains the single integer 12, To 3 appertain the two integers 3, 9, I mo ^j^ To 4 appertain the two integers 5, 8, To 6 appertain the two integers 4, 10, To 12 appertain the four integers 2, 6, 7, 11, Theorem 27. To every positive divisor, t, of <t>(p), there appertain <f>(t) integers 1 with respect to the modulus p. Assume that to every positive divisor, t, of <j>(p), there apper- tains at least one integer, a. We shall show that, if this assump- tion be true, there appertain to t <f>(t) integers; that is, to every positive divisor, t, of <f>(p) there appertain either <f>(t) integers or no integers. Let \J/(t) denote the number of integers apper- taining to t. Each of the integers a° = i,a,a 2 , -,o w 11) is a root of the congruence x* = i, modp, 12) for, if a r be one of these integers, then (ar)*=(a') r =i, modp, since a'=i, modp. The integers 11) are moreover by Th. 23 incongruent each to each, mod p, and, being t in number, are therefore all the roots of 12), since 12) can not have more than t incongruent roots. But every integer appertaining to t must evidently be a root of 12) and we need look therefore only among the integers 11) to find all integers appertaining to t. Let a r be any one of the in- tegers 11). If a r appertain to t, we must have a r ,a 2r , •••,a ( '~ 1)r 1 We, of course, consider only incongruent integers ; see p. 99. THE RATIONAL REALM — CONGRUENCES. IO3 each incongruent to I, mod p. By Th. 24 the necessary and suffi- cient condition for this is ir^to, modt, 13) where i runs through the values 1,2, •••, t — 1. In order now that 13) may hold, we must have r prime to t ; for suppose that the contrary is true and that d is the greatest common divisor of r and t, assuming for convenience d to be positive. We have r=r x d, t = t x d, and, since t x < t and i runs through all values from 1 to t — 1, one of the values of t will be t x and we shall have for this value t x r x d = o> mod t x d; that is, 13) does not hold. But, since i < t, 13) holds whenever r is prime to t. Hence the necessary and sufficient condition that any one, a r , of the integers 1, a, a 2 , •■•, a* -1 shall appertain to t, is that its exponent, r, shall be prime to t. This condition is fulfilled by <f>(t) of these integers, and we have proved therefore that tf,(t) = either </>(/) oro. We shall now prove that the latter case can never occur. We separate the <f>(p) integers of a reduced residue system, mod p, into classes according to the divisor of <f>(p) to which they apper- tain; that is, if t ly t 2 ,---,t n be the positive divisors of <f>(p), we put in one class the \p{t x ) integers of the above system that apper- tain to t ly in another class the if/(t 2 ) integers that appertain to £0, etc. It is evident that no integer can belong to two different classes and that every integer must belong to some one of these classes. The integers of a reduced residue system, mod p, being <f>(p) in number, we have therefore Hh) +Hh) + - +tM=<i>(P). But by Th. 5, <f>(p) taking the place of m, we have <K'i) +<t>(t 2 ) + ••• + <f>(t n ) =<£(/>), 104 THE RATIONAL REALM CONGRUENCES. whence #(«l) +*(*•) H hHtn) =<f>(t 1 ) +<j>(t 2 ) + • • • +<f>(t n ). 14) Since, however, every term in the first member of 14) is equal either to the corresponding term in the second member or o, if even a single term in the first member were o, 14) would not hold. Hence no term in the sum ^(^) + ^(f 2 ) +""* +♦(*») 1S °- Therefore f(t) = <j>(t). § 29. Primitive Roots. An integer, that appertains to the exponent <f>(m) with respect to the modulus m, is said to be a primitive root of m. For example; 2, 6, 7 and 11 appertain, mod 13, to the exponent 0(13), = 12, and are therefore primitive roots of 13. It can be shown that such integers exist only when m = 2, 4, p n or 2p n , where p is an odd prime. 1 We shall discuss however only the case where m is a prime number. It having been proved in Th. 27 that, if p be a prime, there appertain <j>(<j>(p)) integers to the exponent <f>(p), mod p, we see that p has always <f>(<j>(p)) incongruent primitive roots. If r be a primitive root of p, then by Th. 23 the <j>(p) powers of r r,r 2 ,---, r0 (p) form a reduced residue system, mod p. Hence every integer, that is not divisible by p, is congruent to one of these powers of r, mod p. This property, upon which depends the use- fulness of a primitive root, may be used to define it as follows : An integer, a complete system of zvhose pozver residues, mod m, constitute a reduced residue system, mod m, is called a primitive root of m. For example; 2, 2 2 , 2 3 , 2 4 , 2 5 , 2 6 , 2 7 , 2 8 , 2 9 , 2 10 , 2 11 , 2 12 con- stitute a reduced residue system, mod 13. Hence 2 is a primitive root of 13. We shall illustrate the advantage of this representation of a reduced residue system by a second proof of the generalized form of Wilson's theorem (Th. 21). Let p be an odd prime, r a primi- 1 Gauss : Disq. Arith., Arts. 57-93. Dirichlet-Dedekind : §§ 127-131. Bachmann : Elemente der Zahlentheorie, pp. 89-104. Bachmann : Niedere Zahlentheorie, pp. 322-348. Mathews : §§ 19-29. Wertheim : §§ 48-69. THE RATIONAL REALM — CONGRUENCES. 105 tive root of p, and q 1 ,q 2 ,--,q 4 ,<P) any reduced residue system, mod p. Since the integers r,r 2 , -~,r<i> ip} constitute a reduced residue system, mod p, each of the q's must be congruent to some one of these powers of r, mod p ; that is, mod p, where l lt l 2 , — ,J*c« are the numbers 1,2, —,♦(#) in some order. Multiplying these congruences together, we have But r 1+ ( *»=r, mod/', and hence tfi# 2 **•> 2<mp> — r 2 » mod p. 1) We have also r0<p> — 1 = (r^ ( ^/ 2 — 1) (V0W/ 2 + 1) =0, mod p, and hence, since r 0(p)/2 — 1^0, mod/>, r being a primitive root of p, r 0(p)/2 _|_ T = 0j m0( J £ # 2 ) Therefore from 1) and 2) it follows that ftfa •"$♦<#> + 1 ssO* mod/>. When p = 2, this proof does not hold as </>(/>) is then odd. § 30. Indices. If q = r*, mod />, r being a primitive root of /> and i one of the numbers o, 1, •••,<£(/>) — x > * is said to be */*£ i«cte.r 0/ g to the base r, mod p, and we write 1= ind r g, mod />. The subscript r is often omitted, in which case it is understood that all indices are to be taken to a certain given base. The relation of an integer to its index is evidently very similar 106 THE RATIONAL REALM CONGRUENCES. to that of a number to its logarithm and indices play a part in the theory of numbers similar to that of logarithms in arithmetic. It can be easily shown that they obey the following laws : Let p be the modulus, and r a primitive root of p. i. The index of the product of two integers is congruent to the sum of the indices of the factors, mod <f>(p), that is, ind r ab ss ind r a + ind r b, mod <£(/>). This result can evidently be extended to the product of any number of integers ; that is, indr (cMV" a n) = ind r a x + ind r a 2 + ""+ ind r a„, mod <j>(p). ii. The index of the nth power of an integer is congruent to n times the index of the integer, mod <j>(p), n being a positive in- teger; that is, indr a n = n ind r a, mod <f>(p). To prove i, from which ii at once follows, let ind r a = i lt ind r b = i 2 , ind r ab — i. Then assr* 1 , modp, Z? = r f 2, mod />, ab^r 1 , mod p, and hence r 4 = r il+ * 2 , mod p. Therefore by Th. 24 i = i x -f i 2 , mod <f>(p) ; that is indr ab = ind r a + ind r b, mod <f> (p ) . We observe that in every system ind r 1=0. By means of the following tables, we can verify these results and illustrate the use of indices. Table A gives for the modulus 13 the index to the base 2 of each integer of a reduced residue system, and Table B gives the residue corresponding to any index for the same base and modulus. It is evident that two integers congruent to each other, mod p, have the same index in any system of indices, mod p. Jacobi has given in his Canon Arithmeticus, Berlin, 1839, such tables for all primes less than 1000. See also for such tables for all numbers less than 100 that have primitive roots Wertheim, Elemente der Zahlentheorie, also Cahen for list of primitive roots and tables of indices for every prime number less than 200. THE RATIONAL REALM — CONGRUENCES. A. 107 Residue... Index 1 2 1 3 4 4 2 5 9 6 5 7 11 8 1 3 9 8 10 10 11 7 12 6 B. Index Residue ... 1 1 2 2 4 3 8 4 3 5 6 6 12 t\ 8 9 9 5 10 10 11 7 Ex. Using the above tables, where the modulus is 13 and the base 2, we have ind 2 5 = 9, ind 2 9 =8. Therefore ind 2 45 ^ind 2 5 + ind 2 9^ 17, mod 12, and hence ind 2 45 = 5. This result may be verified by observing that 45 = 6, mod 13, whence ind 2 45 s ind 2 6, mod 12 ; that is, ind 2 45= 5. We can pass from a system of indices with base r lf modp, to one with the base r 2 and the same modulus by a process similar to that employed in passing from one system of logarithms to another. Let p be the modulus, a any integer not divisible by p, and f 1 = indr 1 a, i 2 = 'md r a, t='indr 1 *V Then we have a=^rj*, mod/>, and also a = r 2 i2 , mod p. But r 1 ^r 2 i J modp, and hence from 2) and 3) it follows that a = jy'S modp, whence ind r a = ii^ mod <f>(p) ; that is, ind r a = ind rg r i ' ind^a, mod <f>(p). Therefore, to obtain a system of indices to the base r 2 for a given modulus p, from one to the base r x , we have only to multiply each index of the latter system by ind T ^r x and take the smallest positive residue of the products with respect to the modulus <f>(p). If r lf r 2 be any two primitive roots of p, then ind r l r 2 -indr 2 r l ^i, mod<f>(p). This follows at once from 4) by putting a = r 2 . 2) 3) 4) io8 THE RATIONAL REALM — CONGRUENCES. Ex. To obtain for the modulus 13 a system of indices to the base 7 from one to the base 2, we have first to find ind7 2. We have ind 2 7-ind T 2^ 1, mod 12, and from table A ind 2 7= II, whence nind T 2^ 1, mod 12. Therefore ind 7 2 = 11. Multiplying by 11 each index to the base 2 and taking the least positive residues of these products with respect to the modulus 12, we obtain for the modulus 13 the following system of indices to the base 7. Residue... Index I 2 3 4 5 6 7 8 9 10 11 II 8 10 3 7 1 9 4 2 5 12 6 Theorem 28. // ind r a, mod p, be i and d be the greatest com- mon divisor of i and p — I, then a appertains to the exponent {p-i)/d. We have a = r*, mod p. We ask what is the smallest value of m for which a m = r mi =i, mod p. 5) By Th. 24 we must have mi = o, modp — 1, and hence m p-l d = °> m0d -d 6) But i/d is prime to (p — i)/d and (p — i)/d is therefore the smallest value of m greater than zero, that will satisfy 6). Hence (p — i)/d is the smallest value of m that will satisfy 5) ; that is, a appertains, mod p, to the exponent (p — i)/d. Cor. If r be a primitive root of p, then the <f>(p — /) primitive roots of p are those (f>(p — 1) incongruent powers of r whose exponents are prime to p — 1. Ex. One primitive root of 13 is 2. Hence the 4, = 0(12), primitive roots of 13 are 2, 2 5 , 2 7 , 2 U . §31. Solution of Congruences by means of Indices. If we have a table of indices to any base for a given modulus p, we can solve any congruence of the form THE RATIONAL REALM — CONGRUENCES. IO9 ax=b, mod/>, i) where a is not divisible by p ; for from i) it follows that ind a -j-r ind x = ind b, mod <f>(p), which gives indjr==ind& — ind a, mod <f>(p), from which we can determine ind x and then x. Ex. From the congruence 7x^4, mod 13, » we have ind x ^ ind 4 — ind 7^2 — 11^ — 9, mod 12. Hence ind* = 3, and therefore x^8, mod 13. The solution of the congruence ax n ^b, mod/>, 2) where a is not divisible by p, can be reduced by the use of indices to the solution of a congruence of the first degree, mod <f>(p). For from 2) we have ind a -f- n'mdx='mdb, mod <f>(p), and hence nind;r = indfr — ind a, mod <f>(p), 3) that is, a congruence of the first degree in the unknown ind x. By Th. 12 the necessary and sufficient condition that 3) shall be solvable is that indfr — ind a shall be divisible by the greatest common divisor, d, of n and <f>(p). When this condition is sat- isfied 3) gives \d\ values of ind;r, corresponding to which we find \d\ values of x, that satisfy 2) and are incongruent, mod p. In the following examples 2 is understood throughout to be the base of the system of indices employed, tables A and B being used. Ex. 1. From the congruence $x~ ^4, mod 13, we have 7 ind x ^ ind 4 — ind 5^2 — 9 ^ — 7, mod 12. whence, upon removal of the factor '/, that is prime to the modulus 12, we have ind .*•=== — 1, mod 12. ( IIO THE RATIONAL REALM CONGRUENCES. Therefore ind* — n, and *s=7, mod 13. Ex. 2. From the congruence 4* 15 = 5, mod 13, 4) we have 15 ind * see ind 5 — ind 4 e= 9 — 2 ==7, mod 12. The greatest common divisor of 15 and 12 does not divide 7. Hence 4) has no roots. Ex. 3. From the congruence x 9 ==8, mod 13, we have 9 ind x^ ind 8 ==3, mod 12. 5) The greatest common divisor of 9 and 12 is 3 and it divides the second member, 3, of 5). Hence 5) has 3 roots, that we find by the method of Th. 12. From 5) we have 3 ind x ^ 1, mod 4, whence ind* ==3, mod 4, and consequently ind x^= 3, 7, 11, mod 12. Therefore ind* = 3,7, or 11; and x^8, 11, or 7, mod 13. § 32. Binomial Congruences. The subject of power residues and in particular that portion relating to primitive roots may be treated from another point of view, that of the binomial congruence x n — is=o, mod p. 1 1) We see by §25 that all roots of 1) are roots of the congruence <\>{x) =0, mod/>, where <j>{x) is the greatest common divisor, mo&p, of x n — 1 and x?- x — 1. It is easily seen that <f,(x)=x d —i, where d is the positive greatest common divisor of n and p — 1. The congruence x n — 1=0, modp, ^ahen: p. 77. Bachmann : Niedere Zahlentheorie, p. 318. H. J. S. Smith: pp. 140-145. THE RATIONAL REALM CONGRUENCES. Ill has therefore d incongruent roots, that are the roots of x d — i==o, mod/>. 2) We can now confine ourselves to congruences of the form 2), where d is a divisor of p — 1. The roots of 1) fall into two classes, those which satisfy no congruence of the same form and of lower degree, these being called primitive roots, and those which satisfy congruences of this form and of lower degree, these being called imprimi- tive rootg. It is easily seen that every integer that is a root of a con- gruence #* — 1^0, mod/v, 3) where d ± is a divisor of d, is also a root of 2), and conversely that every imprimitive root of 2) is the root of a congruence of the form 3), where d ± is a divisor of d smaller than d. The primitive roots of 2) are evidently, in the language of power residues, those integers that appertain to the exponent d, mod^. They are evidently <j>(d) in number (Th. 2j). The primitive roots of p are the primitive roots of the congruence xp- 1 — 1=0, mod p. The product of any number of roots of 2) is a root of 2) and, in particular, any positive integral power of a root of 2) is a root of 2). If r be any primitive root of 2), then the d roots of 2) are by Th. 23 I, r, r 2 , -", r*- 1 . If a lf a 2 be roots of the congruences x dl — 1=0, mod p, 3) and x* 2 — is==o, mo&p, 4) respectively, then a x a 2 is a root of the congruence j^id, — 1=0, mod p. 5 ) In particular, if a lf a 2 be primitive roots of 3) and 4) respect- 112 THE RATIONAL REALM CONGRUENCES. ively and d 1} d 2 be prime to each other, then a ± a 2 is a primitive root of 5) (Th. 26). The close analogy between the theory of binomial congruences and that of binomial equations will be easily seen. §33. Determination of a Primitive Root of a Given Prime Number. 1 The method, which is due to Gauss, depends upon the deter- mination of a series of integers each of which appertains to a higher exponent with respect to the given prime, p, than any of the preceding ones. In such a series we must evidently reach an integer which appertains to the exponent p — 1, modp; that is, which is a primitive root of p. Take any positive integer, a lf less than p and greater than 1, and form a complete system of its power residues, modp. Let us suppose that a x appertains to the exponent t lt modp. If t x = p — 1, then a ± is the primitive root required. If t 1 =^=p — 1, it is evident that none of the power residues of a x Can be a primitive root of p, for they are the roots of the congruence jfr — 1 = o, mod p, 1 ) and hence appertain, modp, to exponents not greater than t v Suppose that t 1 =^=p — 1. We proceed to determine an integer appertaining, modp, to an exponent greater than t x . Select any positive integer, a 2 , less than p and not contained among the power residues of a lt modp, and form a complete system of its power residues, mod p. Let t 2 be the exponent to which a 2 apper- tains, modp. If t 2 —p — 1, a 2 is a primitive root of p and the problem is solved. Suppose that t 2 =^=p — 1; then t 2 can not be a divisor of t ti for a 2 would in that case be a root of the con- gruence 1) and hence a power residue of a lf modp, which is contrary to our hypothesis. If t 2 be a multiple of ^ but =%=p — 1, we have found an integer, 1 Gauss : Disq. Arith., Art. 73 Ca'.ien : pp. 90-95. Mathews : pp. 20-22. H. J. S. Smith : pp. 49-54. THE RATIONAL REALM CONGRUENCES. II3 a 2 , appertaining to a higher exponent than a lt modp, although not a primitive root of p. We then select a positive integer less than p and not contained among the power residues of a 2 , form its power residues, modp, and proceed as before. Suppose, how- ever, that t 2 is not a multiple of t ls and let m be the least common multiple of t x and t 2 . It is evident that m is greater than t lf since t 2 is, not a divisor of t v We shall show how to determine an integer appertaining to the exponent m, mod p. We first resolve m into two factors, m lf tn 2 , prime to each other and divisors of t x and t 2 respectively. This may be accom- plished as follows. Let p x be a prime that occurs to the power e 1 as a factor of t x and to the power e 2 as a factor of t 2 . We take p x ei as a factor of m lf or p x e2 as a factor of w 2 , according as e x is greater or less than e 2 . If e 1 = e 2 , then /^i may be taken as a factor of either m t or w 2 . We have then m = m 1 m 2 = t 1 /d 1 -t 2 /d 2 , where d ls d 2 are respectively the product of primes that occur in the case of d x to a lower power in t x than in t 2 , and in the case of d 2 to a lower power in f 2 than in t ± . Consider now the residues, modp, of a/ 1 , and a/ 2 . These integers appertain respectively to the exponents tjd x , tjd 2 , that are prime to each other. Hence their product a x di a 2 d2 appertains to the exponent m, that is the product of these exponents (Th. 26). Ex. To find a primitive root of 157. The power residues of 2, mod 157, are 2, 4, 8, 16, 32, 64, 128, 99, 4i, 82, 7, 14, 28, 56, 112, 67, 134, in, 65, 130, 103, 49, 98, 39, 78, 156, — 2, — 4, — 8, -16, — 32, — 64, —128, — 99, — 4i, -82, — 7, — 14, — 28, -56,- — 112, — 67, —134,- — in, -65, — 130, -103, — 49, -98, — 39, -78, -iS6asi. The work is shortened by observing that the residue of 2 28 is — 1, and consequently the remaining 26 residues are the negatives of the first 26. We see that 2 appertains to the exponent 52, mod 157. The integer 3, not being contained among the residues of 2, we form its power residues, mod 157, and find that it appertains to the exponent 78. 8 114 THE RATIONAL REALM CONGRUENCES. We have 52 = 2 2 -i3, and 78 = 2-3-13. The least common multiple of 52 and 78 is 156, that can be resolved into two factors prime to each other and divisors of 52 and 78 respectively. Thus ^^x'-^-^^X^. 13 2 13 2 The integers 2 13 and 3 2 appertain to the exponents — and — respectively, 13 2 and hence their product 2 13 3 2 appertains to the exponent 156; that is, 2 13 3 2 is a primitive root of 157. But we have seen that 2 13 ==28, mod 157. Hence 2 13 -3 2 ==28-9 = 252==55, mod 157. We have therefore 55 as a primitive root of 157. We could have resolved 156 in another way, since 13 occurs to the same power in 52 and 78. Thus I56 = ^I3 x i^_L3 = 5£ x Z? J I ^ 2 • 13 I /X 26 Then 2 and 3 26 appertain to the exponents 52 and 3 respectively, and their product 2-3 M appertains to the exponent 156; that is, 2-3 28 is a primitive root of 157. We have 2-3*^ 2- 144 ^288^ 131, mod 157, and hence 131 is a primitive root of 157. For this example and a table of the power residues of 55, mod 157, see Cahen : pp. 92, 93. §34. The Congruence x H E=b, mod p. Euler's Criterion. The congruence C^j^essfr^ modp, where a x is not divisible by p, can always be reduced to the form x n = b, modp, and in this form it has a special interest. In what follows we consider b^o, mod p. From what has been said in §31, the truth of the following theorem is at once evident. Theorem 29. The, necessary and sufficient condition that the congruence x n ^=b, modp, 1) THE RATIONAL REALM CONGRUENCES. I I 5 shall be solvable, is that ind b shall be divisible by the greatest common divisor, d, of n and <f>(p) ; this condition being satisfied the congruence has exactly \d\ incongruent roots. See § 31, Ex. 3. Since ind"& varies with the primitive root taken as base of the system of indices used, this condition for the solvability of 1) appears to depend upon the primitive root selected. It is evident, however, that in reality the solvability of i) is in no way dependent upon this selection, and it must be possible therefore to find a criterion for the solvability of this congruence that is independent of indices. Such a criterion is that first given by Euler and known as Euler's criterion. It is contained in the following theorem. Theorem 30. // d be the positive greatest common divisor of n and <f>(p), the necessary and sufficient condition that the congruence x n = b, mod p, 2) shall be solvable is b4>w/ d =i, mod p. j) This condition being satisfied, the congruence has exactly d incon- gruent roots. Let r be any primitive root of p, and let indr b = c. Suppose 2) to be solvable, then c is divisible by d. Let c = md. Then b^r md , mo&p, and btw/d^rm-tw, modp, whence b<pw/d= I} mo d/>. Therefore 3) is a necessary condition for the solvability of 2). Conversely, if b satisfy 3), the index of b in every system of indices, mod p, must be divisible by d ; for, if b==r c , modp, then b<t>(pWd = r c<f>( P )/d } m od/>, Il6 THE RATIONAL REALM CONGRUENCES. and hence r c0d»/tf== I? modp. Since r, being a primitive root of p, appertains to the exponent <f>(p), c<f>(p)/d must be divisible by <\>{p). Therefore c/d is an integer ; that is, c is divisible by d. Hence 3) is a sufficient as well as necessary condition for the solvability of 2). That the congruence when solvable has d roots is evident from the preceding paragraph. All incongruent integers b, for which the congruence 2) is solvable may be obtained by observing that they are the roots of the congruence ,r*<9>/*sBi, mod p. 4) This congruence has <f>(p)/d incongruent roots, since <f>(p)/d is a divisor of <f>(p). These roots are the incongruent, modp, values of b for which 2) is solvable. Such numbers congruent to the nth power of an integer, modp, are called the «-ic resi- dues of p, and we have the following theorem. Theorem 31. The number of incongruent n-ic residues, mod p, is <f>(p)/d, where d is the positive greatest common divisor of n and <f>(p), and these residues are the rodts of the congruence x<t>w/ d ==i, modp. Thus, if p = 7, we have for 11 = 2, 3 incongruent quadratic residues of 7, w = 3, 2 incongruent cubic residues of 7, n = 4, 3 incongruent biquadratic residues of 7, m = 5,6 incongruent quintic residues of 7, n = 6, 1 incongruent sextic. residue of 7, and so on. We may obtain the above results and also the residues them- selves by raising each number of a reduced residue system, mod p, to the nth power and determining the number of the reduced residue system to which each of these nth powers is congruent, THE RATIONAL REALM — CONGRUENCES. II7 mod p. Thus for p = 7, we take as a reduced residue system 1, 2, 3, 4, 5, 6, and have for n = 2, i 2 = i, 2 2 = 4, 3 2 = 2, 4 2 = 2, 5 2 = 4 , 6 2 ^i/ w = 3, i 3 = i, 2 3 e==i, 3 3 = 6, 4 3 =i, 5 3 = 6, 6 3 = 6, n = 4, i 4 =i, 2 4 = 2, 3 4 = 4, 4 4 = 4, 5 4 = 2 > 6 4 =i, [.mod 7. w = 5, i 5 =i, 2 5 = 4, 3 5 = 5. 4 5 = 2, 5 5 = 3, 6 5 = 6, n = 6, i'bbi, 2 6 =i, 3 6 =i, 4 6 =i> 5 6 = i> 6 6 =i, Hence the incongruent quadratic residues of 7 are 1, 2 and 4, the cubic residues 1 and 6, the biquadratic residues 1, 2 and 4, the quintic residues 1, 2, 3, 4, 5 and 6, the sextic residue 1. An integer is therefore a quadratic residue of 7 when and only when it is congruent to one of the integers 1, 2, 4, mod 7, and likewise for the other values of n. In the next chapter we shall discuss fully the subject of quad- ratic residues. Investigations concerning the properties of cubic and biquad- ratic residues have led to important developments in the theory of numbers, that will be noticed later. Examples. 1. Show that ,r 13 — x is divisible by 2730, x being any integer. 2. If x be a prime greater than 13, x 12 — 1 is divisible by 21840. p(p-i) 3. If p be a prime and a prime to p, then either a 2 — 1 or a 2 + 1 is divisible by p 2 . 4. No number of the form m 4 -f- 4 except 5 is prime. 5. The product of numbers of the form mx + 1 is a number of the same form. 6. The cube of any integer not divisible by 3 is congruent to ± 1, mod 9. 7. Solve the congruences a) x 3 — &r+ 1=0, mod 5. b) x 4 + 6x 3 — 8x 2 + lyc + 5 = o, mod 7. c) x*-\-2x 3 — i3.F 2 -f S x + x 3 = > m od 11. 8. The congruence mod 15. I 1 8 THE RATIONAL REALM CONGRUENCES. 8.r 5 + 4-r 4 — 3,i- 3 -f- 3X 2 + 3^ + 6 = o, mod 7, has a multiple root; solve the congruence. 9. Solve the system of congruences 3* — 43'+ 52 — 911 = 1 2* + $y+ 4^ + 5^= 8 ■*" + 53' + 6z-{-2u= 1 7 X — 33' — I02 + 2U ■■ I0 10. Solve the congruence a- 5 — &r 4 + 5jt 3 — 5-r 2 + 4.1- -j- 3 = o, mod 27. 11. Solve the congruence x 5 — 6.r 4 + 8a- 3 — 4* 2 + jx -f 2 = 0, mod 20. 12. Prove Th. 30 without the use of indices. 13. Find the prime polynomials of the third degree, mod 5. 14. If a appertain to the exponent t a , mo&p, then 1 -f a + a 2 -f • • • + a**- 1 s== o, mod p, (Gauss: Disq. Arith., Art. 79.) 15. The product of all incongruent primitive roots, mo&p, is congruent to 1, mod/?, except when p = 3. {Ibid.: Art. 80.) 16. If r lt r 2 , •••,r^ m) be a reduced residue system, modw, then all primes are contained in the forms km + r lf km + r 2 , • • • , km + r Um) . 17. If p be a prime of the form \n — 1 and a appertain, mod/>, to the exponent (p — 1)/2, then — a is primitive root of p. 18. Use theorem in Ex. 17 to determine a primitive root of 191. (Cahen: p. 94.) 19. Determine a primitive root of 73 (Gauss: Disq. Arith., Art. 74), also one of 97 (Mathews : p. 20). 20. If p be a prime and r x ,r 2 , •••,^ (P) a reduced residue sys- tem, mod/>, every rational integral symmetric function of the r's, whose degree is not a multiple of (f>(p), is divisible by p. (Cahen: p. 109.) 21. Solve the congruences a) x 20 = 3, mod 13. b) ,r 9 = io, mod 13. CHAPTER IV. The Rational Realm, quadratic residues. 1 § i. The General Congruence of the Second Degree with One Unknown. The most general congruence of the second degree with one unknown has the form ax 2 + bx + c — °> m °d m. I ) We have seen (Chap. Ill, $2j) that the solution of i) when m is a composite number can be reduced to the solution of a system of congruences of the same form but with prime moduli. We shall therefore confine ourselves to the case in which m is a prime number, p, and furthermore, since for p = 2 the congruence is easily solvable by trial, we shall suppose p odd. We consider then the congruence * ax 2 + bx -\- c = o, mod p, 2) where a is not divisible by the odd prime p, for if it were, the con- gruence would not be of the second degree. Multiplying 2) by the reciprocal, a t , mod p, of a, we obtain the congruence x 2 + a x bx -\- a x c == o, mod p. 3) If now the coefficient of x in 3) be not even, we make it so by putting aj? + P for (hf>. Having done this, if necessary, 3) is transformed into the equivalent congruence x 2 + 2b t x -f- c x =5 o, mod p. 4) Adding b ± 2 to both members of 4), we obtain (x-{-b 1 ) 2 = b 1 2 — c lt modp, or putting x + b x = s, mod />, 5) & x 2 — c x ^d, mod/', 1 Gauss: Disq. Arith., pp. 73-119. Wertheim: pp. 170-236. Cahen: pp. 113-143. Bachmann: Niedere Zahlentheorie : pp. 180-317. Dirichlet- Dedekind: pp. 75-127. 119 120 QUADRATIC RESIDUES. we see that the solution of 2) can be reduced to the solution of a binominal congruence z-^d, mod/>. 6) If d^o, mod/0 7) the congruence 6) has either no roots or two incongrnent roots, for if r be a root, then — r is also a root, and if r== — r, modp, then 2r==o, mo&p, and hence r = o, mod/>, which is impossible from 7). The solutions of 4), or what is the same thing 3), being con- nected with those of 6) by the relation 5), we see that 4) has two incongruent roots or no roots according as 6) has two incon- gruent or no roots. If d = o, mo&p, then 6) has the two equal roots £ = 0, modp, and 4) has the two equal roots 1 #h== — b 19 modp, 1 x 2 + 2b ± x + C\ being a perfect square, mod p. The solutions in the case of equal roots being obvious, we shall exclude this case and confine ourselves therefore to the consideration of binomial congruences of the form 6), where c?HJ=o, mod p. The analogy shown here between quadratic equations and congruences of the same degree with prime modulus should be noticed, the vanishing of the discriminant b 2 — \ac of ax 2 -\-bx-{- c being in the one case the condition that the equation ax 1 + bx 4- c = o, shall have equal roots, and the divisibility of b 2 — 4ac by the modulus being in the other case the condition that the congruence 1 Wertheim: p. 170. ♦ QUADRATIC RESIDUES. 121 ax 2 + bx -f- c ra o, mod />, shall have equal roots. Ex. Let 5* 2 — \ix — 12 = 0, mod 23, be the proposed congruence. Multiplying it by 14, the reciprocal, mod 23, of 5, we obtain the equivalent congruence. 70X 2 — 154* — 168 ==0, mod 23, or x 2 — 16* — 7 == o, mod 23, or (x — 8) 2 ==2, mod 23. Putting x — 8 ==2, mod 23, 8) we have ^^2, mod 23, which has the roots z as 5 or — 5, mod 23. These substituted in 8) give the two roots of the original congruence x^i$ or 3, mod 23. § 2. Quadratic Residues and Non-residues. An integer, a, prime to the modulus m. is said to be a quadratic residue or non-residue of m, according as the congruence „v 2 ==a, mod wi, has or has not roots; that is, a is said to be a quadratic residue of m, if it be a residue, mod m, of some square number, and a quad- ratic non-residue of m, if it be a residue, mod m, of no square number. Ex. 1. The congruence x^^2, mod 7, has the roots 3 and — 3 ; hence 2 is a quadratic residue of 7. Ex. 2. The congruence x 2 ^s, mod 7, has no roots, as may be seen by trying the integers — 3, — 2, — 1, o, 1, 2, 3 (also see Chap. Ill, § 34) ; hence 5 is a quadratic non-residue of 7. If there be no danger of misunderstanding, the word quadratic is omitted. The behavior of the integer a in this relation is called its quadratic character with respect to the modulus m. It is evi- dent that all integers belonging to the same residue class, mod m, have the same quadratic character with respect to m. We have now two principal questions to answer concerning the congruence x 2 ?==a, modw. 122 QUADRATIC RESIDUES. I. What integers are quadratic residues of a given modulus m? II. Of what moduli is a given integer, a, a quadratic residue? We shall confine ourselves now to the case in which, m is a prime, p. Furthermore, we may suppose p to be odd, since the case p = 2 is at once disposed of by observing that all odd integers are quadratic residues of 2, and all even integers, being not prime to 2, are excluded from consideration. For convenience, we also suppose p to be positive. We have as a special case of Th. 30, Chap. Ill, the following : Eule/s Criterion. Theorem i. The necessary and sufficient condition that a shall be a quadratic residue of p; that is, that the congruence x 2 = a, mod/>, shall have roots, is a (p_1)/2 =i, mod p. - Cor. 1. The integer a is a quadratic residue or non-residue of p according as we have a <p-i>/* == 1, or — 1, mod p ; for since aP~ x — 1=0, mod p, then ( a W 2 — 1 ) (a^/ 2 + 1 ) = o, mod p ; whence it follows that either a (p-D/2_ I == 0> mod/), or a ( P -1)/2 _|_ 1 sb o, mod p. -Therefore if a ( P- 1)/2 == 1, mod p, a is a quadratic residue of p, and if a ( P- 1) / 2 == — i } m od p, a is a quadratic non-residue of p. Cor. 2. The product of two quadratic residues or of two quad- ratic non-residues of p is a quadratic residue of p, and the product of a quadratic residue and a quadratic non-residue of p is a quad- ratic non-residue of p. Let a lf a 2 be quadratic residues, and a 3 , a 4 quadratic non-residues of p. Then since a 1 ( p _1) / 2 ^ 1, modp, QUADRATIC RESIDUES. I 23 and a*<**>/**arl, mod p, it follows that . O^) (p ~ 1)/2 = I, mod p. Hence a x a 2 is a quadratic residue of />. Since ' fl3<p-i)/2=_ 1, mod/', and a/p- 1 )/ 2 ^— 1, mod/>, it follows that Os^) (p ~ 1)/2 ■■ I, mod p. Hence a 3 a 4 is a quadratic residue of />. Since ^^/'a I, mod/>, and ^OHiVS = — i, mod />, it follows that O^g) **>/*■»— i, mod p. Hence a x a 3 is a quadratic non-residue of />. From Cor. 2 follows at once : Cor. 3. The product of several integers is a quadratic residue or non-residue of p, according as an even or odd number of the integers are quadratic non-residues of p. It is therefore only necessary to be able to determine the quad- ratic character of all prime numbers with respect to any modulus p. Ex. 1. **eB3, mod 13. 1) We have 3 (13_1) /2 = 3 6 = I, mod 13. Hence 3 is a quadratic residue of 13, the roots of 1) being 4 and — 4. Ex. 2. ;Te=7, mod 13. Hence 7 is a quadratic non-residue of 13. We can verify the result by substituting the numbers, ±1, ±2, ±3, ± 4, ■ ± 5, ±6, which give I £7 9¥k7 25^7 I modl3 . 4=^7 16^7 36=£7 j This also follows from the fact that ind 2 7, mod 13, is not divisible by 2. Ex. 3. Since 21 = 3-7 and 3 is a residue of 13, and 7 a non-residue of 13, 21 is by Cor. 2 a non- residue of 13, which is verified by 2I (13- 1 ) / 2_ ( _ 5) 6 ) modl3> ^((-5) 2 ) 3 ^(-i) 3 ^-i. mod 13. 124 QUADRATIC RESIDUES. § 3.? Determination of the Quadratic Residues and Non- residues of a Given Odd Prime Modulus. Theorem 2. // p be an odd prime, one half the integers of a reduced residue system, mod p, are quadratic residues of p } and the other half non-residues. First Proof: Take as a reduced residue system, mod p, the integers p-i p- 3 _ 2 _ I2 . P-I P-* _x 2 2 2 2 ' The squares of the integers 2 2 are incongruent each to each, mod p, for if (p — r)/2 and (p — s)/2 be any two of these integers, r and s being integers of the series 1,3, •••,/> — 2, and unequal, and m. (^r + ti)(^-^)-o.„od A whence either 1- = o, mod p, 3) 2 2 or P^ll — t—L m o, mod p y 4) 2 2 Both 3) and 4) are, however, impossible, since (p — r)/2 and (p — s)/2 are unequal and both positive and less than p/2. The squares of the -J (p — 1) integers 2) give, therefore, -J (p — 1) incongruent residues, mod p, and these are all the incongruent quadratic residues of p } for the squares of the remaining integers of 1) give evidently the same residues. Hence the theorem. QUADRATIC RESIDUES. 125 Second Proof: Let r be a primitive root of p. Then r,r 2 , '■-,r i , •••,rP- 1 is a reduced residue system, mod p. From Chap. Ill, Th. 29, it follows at once that every power of r with an even exponent is a residue of p, and every power of r with an odd exponent is a non-residue. Hence there are J(/> — 1) residues of p and \{p — 1) non- residues of p. We can express this also by saying that those of the integers of a reduced residue system which have even indices are residues of p, while those which have odd indices are non-residues. The residues of any prime for which we have a table of indices can evidently be easily thus determined. Th. 1, Cor. 2, can be deduced from the second proof given above in a very elegant manner ; for if a^=a x a 2 ••• a n , then ind a = ind a x + ind a 2 -f- • • • -f- ind a n , mod <j>(p), and hence, since <f>(p) is even, ind a is odd or even according as ind a x -f- ind a 2 -f- • • • -j- ind a n is odd or even. But ind a x -j- ind a 2 -f- • • • -f- ind a n , and hence ind a, is odd or even according as an odd or even number of the indices of a x ,a 2 , ••- J (in are odd. Hence a is a quadratic residue or non-residue of p according as an even or odd number of its factors a^,a 2 , "-,a n are quadratic non-residues of p. We can now answer fully the first of our two questions con- cerning the congruence # 2 = a, modpj where p is an odd prime ; for suppose that we have any reduced residue system, mod p, and that those residues of this system which are quadratic residues of p, are r lr r 2 , ••-,r^( i , ) and those which are quadratic non-residues of p are n 1} n 2 , '",n^( P ), this having been determined by any of the methods given above. Then all those and only those integers included in the forms kp + r lf kp + r 2 ,--., kp + r mp) 126 QUADRATIC RESIDUES. are quadratic residues of p, and all and only those integers included in the forms kp + n lt kp -f- n 2 , •••, kp + w^( P ) \ are quadratic non-residues of p, k taking all integral values. \ Ex. i. Let p = 17, and take as a reduced residue system, — 8, —7, —6, —5, —4, —3, —2, —1, 1, 2, 3, 4, 5, 6, 7, 8 We have (±i) 2 =l, (±3) a = 9, (±5)'— 81 (±7)^15,-1 ^ od (±2) 2 = 4, (±4) ! =i6, (±6) 2 = 2, (±8) : .5,-1 13./ Hence 1, 2, 4, 8, 9, 13, 15, 16 are the incongruent quadratic residues of 17, and all those and only those integers which are included in the forms 17k + 1, 17k + 2, 17& + 4, I7& + 8, I7& + 9, 17^ + 13, I7H- 15, 17^ + 16, are quadratic residues of 17. The incongruent quadratic non-residues of 17 are 3, 5, 6, 7, 10, 11, 12, 14, and hence all and only those integers which are included in the forms *7* + 3, Wk + 5, vjk + 6, 176 + 7, 17* + 10, 17k -f 11, 17& + 12, 17& + 14, are quadratic non-residues of 17. Ex. 2. Let p as 13. From table A, Chap. IV, § 30, we see that the indices of 1, 3, 4, 9, 10 and 12 are even, and the indices of 2, 5, 6, 7, 8 and 11 are odd. Hence 1, 3, 4, 9, 10 and 12 are the incongruent quadratic residues of 13, and 2, 5, 6, 7, 8, and 11 are the incongruent quadratic non-residues of 13. We see then, as above, that the quadratic residues of 13 are integers of the forms I3& + Xi I3& + 3, Uk + 4, 13^ + 9, ]3k + 10, 13^ -f- 12, and the quadratic non-residues of 13 of the forms 13& + 2, I3& + 5, 13^ + 6, i3& + 7> 13^+8, 13& + H. We have now answered fully the first question concerning the congruence x 2 = a, mod p ; that is, we are able, as shown in the two examples above, to give for any value of p a finite system of forms, kp -f- r, where r is a known integer and k any integer, such that all and only those integers obtained from these forms by letting k take all integral values, are quadratic residues of p. A similar series of forms may, as was shown above, be given for the non-residues of p. QUADRATIC RESIDUES. \2J Before considering the second question, that is, of what odd prime moduli is a a quadratic residue, we shall introduce a sym- bolic notation which will greatly simplify the discussion. § 4. Legendre's Symbol. The quadratic character of an integer a with respect to a prime p, can be expressed in a very convenient manner by means of the following symbol introduced by Legendre. Let (a/p) denote -f- 1 or — 1, according as a is a quadratic residue or non-residue of p ; that is, (a/p) = i denotes that a is a quadratic residue of p and (a/p) = — 1 denotes that a is a quadratic non-residue of p. In what follows, p is assumed firs t of all to be odd , and secondly , f or the sake__ of convenience, posi^ tive. This last assumption is not necessary, but simply to avoid the trouble of writing \p\ when the absolute value of p is to be taken. Combining this with Euler's criterion, we see that G)-«- mod/, expresses the quadratic character of a, with respect to p. From Th. 1, Cor. 3, it is evident that (^)-(l)G)-G) If a= b y mod /, then Also G)-G> denotes that the quadratic character of a with respect to p is the same as the quadratic character of b with respect to p, and G)~GM*)G)~ ■ 128 QUADRATIC RESIDUES. denotes that the quadratic character of a with respect to p is opposite to the quadratic character of b with respect to p. If a = k 2 a ly then since (k 2 /p) = i, (>)-(') (5) -(5> In determining the value of (a/p) we may therefore suppose all square factors to have been removed from a. § 5. Determination of the Odd Prime Moduli of which a Given Integer is a Quadratic Residue. To answer the second question : of what odd prime moduli is a a quadratic residue, of what a non-residue, we notice first that if a=±q 1 q 2 ••• q n , where q x ,qo,--,q n are the positive prime factors of a we have Hence (a/p) = 1 or — 1 according as an even or an odd number of the symbols (± i/p), (qi/p), '", (qn/p) have the value — 1 ; that is, a will be a quadratic residue of all primes of which an even number or none of the factors ± i,q Xi '-,q n are non-residues. To determine for what values of p the value of (a/p) is 1, for what — 1, it is therefore only necessary to determine for what values of p the value of each of the symbols in the second member of 1 ) is + 1, for what — 1. The problem may be reduced there- fore to the following three simpler ones : To determine 1. Of what odd prime moduli — 1 is a quadratic residue? 2. Of what odd prime moduli 2 is a quadratic residue ? 3. Of what odd prime moduli is another positive odd 1 prime a quadratic residue? § 6. Prime Moduli of which — 1 is a Quadratic Residue. By trial — 1 is seen to be a residue of 5, 13, 17, 29 and a non- residue of 3, 7, 11, 19, 23, and we are led by induction to the fol- lowing theorem: 1 Primary prime. See p. 193. QUADRATIC RESIDUES. I 29 Theorem 3. The unit — 1 is a quadratic residue of all positive primes of the form 4n + 1 and a quadratic non-residue of all positive primes of the form pi — i. 2 We have (§4) ^J^m{^lp t mod A whence, since ( — 1 ) ( p- X) / 2 = 1 or — 1, (¥)-< p-i i) r - Now p has either the form qn -f- 1 or 4^ — 1, and it is easily seen that when p=4n + 1, (—i)<p-v/ 2 = i, aHl^s 4/u *' and when . p = pi — 1, (— t)(p- 1 >/ 2 = — ^ Therefore ( — — J = 1 when p = \n + 1 , and I J= — 1 when p = ^n — I. Ex. 1. We have ( — 1/13) = 1 since 13 = 4-3 + 1; that is, the con- gruence x*?= — 1, mod 13, has roots. These roots are easily seen to be 5 and — 5. Ex.2. We have ( — 1/23)= — 1, since 23 = 4-6 — 1; that is, the congruence x 2 ^ — 1, mod 23, has no roots; a result easily verified. § 7. Determination of a root of the congruence x 2 == — 1, mod p, (p = 4n + 1) by means of Wilson's Theorem. Write down the following congruences, which are evidently true : 2n + 1 = — 2n, mod p, 2n -f- 2 =s — {211 — 1 ) , mod p, 2n + 3= — (2n — 2), modp, 4n = — 1, modp, 2 First given by Fermat ; first proved by Euler. 9 I3O QUADRATIC RESIDUES. and the identical congruence (2ft) != (2^) !, mod p. Multiplying these congruences together, we obtain ( 4 n)!==(— i) 2w [(2ft) !] 2 , modp, or (/- i)|.»J;p=iJ : !J l mod/, But by Wilson's Theorem (p — i ) ! s== — i, mod p, whence {- ) 1 1 = — i, mod p y and therefore * m I- ) !, mod/, is a root of x 2 == — I, mod p. Ex. By the above theorem the congruence x 2 ^ — i, mod 13, has a root x == ( 3 ~ J ! ^= 6 ! ^= 5, mod 13 ; that is, 5 2 == — 1, mod 13. § 8. Gauss's Lemma. The following theorem known as Gauss's Lemma, will enable us to determine (2/p) and (q/p). Theorem 4. // m be any integer not divisible by p and if among the residues of smallest absolute value, mod p, of the products im, 2m, 3m, ■••, %(p — i)m, there be an even number of negative ones, m is a quadratic residue of p, if an odd number, m is a quadratic non-residue ; that is, if fx be the number of nega^ tive residues, (m/p) — ( — /)**. We shall illustrate the content of this theorem by a numerical example. Let /> = I3 and w = 3. The residues of smallest absolute value, mod 13, of the integers 3, 6, 9, 12, 15, 18 are 3, 6, —4,-1, 2, 5, QUADRATIC RESIDUES. I3I two of which are negative. Hence 3 is a residue of 13 ; that is, This is seen to be true since the congruence * 2 e=3, mod 13, has the roots 4 and — 4. To prove the theorem we proceed as follows : Since m is prime to p, the (p — 1)/2 multiples of m P— l X IW, 2m,-'./- m 1) 2 are incongruent each to each, mod p. Their residues of smallest absolute value, mod p, are therefore different integers of the system p — 1 p — 3 p — 3 p — I 2 2 2 2 Those which are positive and belong therefore to the system I, 2, -,^-— -, 2) 2 we shall denote by b lt b 2 , ---,b^. Those which are negative, and belong therefore to the system _, _ 2 ... -P^l 1 > ^1 » > 2 we shall denote by — a x , — a 2 ,--, — a M . Evidently a^, a 2 , - • • , a^ belong to the system 2). Moreover 2 We shall now prove that a^,a 2 , •■•,a IJi ,b 1 ,b 2 , '■•,b x are the integers /— I I32 QUADRATIC RESIDUES. in some order. To do this it will be sufficient to show that no two of these integers are congruent to each other, mod p. It is evident that no two a's are congruent to each other, mod p, and the same is true of the b's. Also no a is congruent to a b, mod p. For if di = bj, modp, and if Km and km be the integers of 1), of which — a-i and bj are the residues of smallest absolute value, mod p, then — hm = km, modp, and hence (h -f- k)m = o, mod p, which is impossible, for m being prime to p, and h and k both positive and < p/2, neither of the factors m or h~\- k is divisible by p. Therefore the (p — i)/2 integers, a lt a 2 , -••,a tL ,b 1 , b 2 , •••,bx, are incongruent each to each, mod p, and being, moreover, all posi- tive and < p/2, must be the integers t*St=± 1,2, , 2 in some order. Since — a x , — a 2 , •••, — a ,b lf b 2 ,'-,b\ are residues of /— l itn, 2m, 3«f, ••• , m, mod /, we have /—I ^ 1 -2.» -^— m f s(-i)^...^ 1 ..^ Al modA whence, since / — 1 and this product is prime to />, we have w 2 ■■■(■— l)*, mod/. QUADRATIC RESIDUES. I 33 But (")- p-1 m 2 , mod p, and (- i) M = 1 or— 1 Therefore (?)-<->• We call /x Gauss's Characteristic. § 9. Prime Moduli of which 2 is a Quadratic Residue. We see by any one of the several methods given, that 2 is a residue of the primes 7, 17, 23, 31, 41, 47, which are of the form Sn zb 1, and a non-residue of the primes 3, 5, 11, 13, 19, 29, 37, which are of the form 8w ± 3. Now every odd prime is of the one or the other of these forms, and the truth of the following theorem seems at once probable. Theorem 4. The integer 2 is a quadratic residue of all primes of the form 8n ± 1, and a quadratic non-residue of all primes of the form 8n± 3. 1 From Gauss's Lemma we have (0- where p is an odd prime, and /* is the number of the integers 2,4,6,---,/> — 1, 1) whose residues of least absolute value, mod p, are negative. To determine when ft is even and when odd we notice that these fi integers are those greater than pi 2. If we suppose the series 1) to be formed by continued subtraction of 2 from p — 1 and write it in the form P— I,/> — 3,"-,p — I— 2(fX— l),p— I— 2fi, •••,4,2, 1 First given by Fermat ; first proved by Lagrange. 134 QUADRATIC RESIDUES. we see that, since there are /x of its terms, beginning with p — I and going backwards, whose residues of least absolute value, mod p, are negative, the smallest one of these terms will be P — i— 2(/x— i). The greatest term whose residue of least absolute value, mod p, is positive is therefore p — I — 2ti. Hence we have p-l -2(/i- i)>|>^_i_2/v 2) From 2) we obtain 4 4 and therefore /* is the greatest integer contained in the fraction (/> + 2)/4. Hence we have, when p = Sn ± i, fi = 2n, and when p = Sn -fc 3, fi = 2n ± 1 ; that is, fi is even when p has the form Sn ± 1, and odd when p has the form 8w ± 3. Hence ( -7 J = i, when/> = 8« ± 1, and f -J==— 1, when/> = 8;i:±3, and the theorem is proved. We can express this result very conveniently in the following manner. We observe that £2 j when p = Sn ± 1, — — == 8n 2 ± 2n Q £2 I and when p = Sn ± 3, —5 — = Sn 2 ±6n + 1 ; o />2 j that is, when p = Sn ± 1, ^ is even, QUADRATIC RESIDUES. 135 and when Hence. Ex. 1. We have p = 8n± 3, is odd. GH-"' 172-1 1 8 . Therefore 2 is a quadratic residue of 17. f v 11 2 -1 Ex.2. We have (_) — (_!) 8 = (_ I )i5__ I> Therefore 2 is a quadratic non-residue of 11. § 10. Law of Reciprocity for Quadratic Residues. It remains now to answer the question : of what odd primes is a positive odd prime q a residue, of what a non-residue? This is answered by means of a theorem which expresses the quadratic character of q with respect to p in terms of the quadratic character of p with respect to q ; thus making the answer depend upon that to our first question, § 2. This theorem, which Gauss has called the " Gem of the Higher Arithmetic," is known as the " Law of Reciprocity of Quadratic Residues," or more briefly as the " Quadratic Reciprocity Law." It is the following: Theorem 5. Law of Reciprocity of Quadratic Residues. 1 If p and q be two different positive odd primes, the quadratic char- acter of q with respect to p is the same as or different from the quadratic character of p with respect to q, according as at least one of the primes is of the form 4n -f- 1, or both are of the form pi — 1; that is, if p = 4h -f- 1 and q = ^k -f- 1, or ^ = 4/i-|-i and g = 4^ — 1, p = ^h — 1 and q = 4k-{-i, or while if p = ^h — 1 and q = $k — 1, ( — (i) a)- 1. x See Bachmann: Niedere Zahlentheorie, pp. 194-318, for a very full discussion of this theorem, a list of all proofs being given. . I36 QUADRATIC RESIDUES. This theorem can be expressed in a very elegant form, if we observe that the expression (p — i)/2-(q — 1)/2 is even when one or both of the primes are of the form 4ft -\- 1, but odd when both are of the form qn — 1. We have, therefore, $(*)-<->?*■ qf \p The proof which follows is due to Pfarrer Zeller, 2 and depends solely on Gauss's Lemma. We have by Gauss's Lemma (!)=(-)-. J where /a is the number of the products \q,2q,--/-^—q, 1) whose residues of least absolute value, mod p, are negative ; likewise where v is the number of the products ip,2p,---, q -^p, 2) whose residues of least absolute value, mod q, are negative. Hence (J)^-*- 1 ^ The problem is therefore resolved into the determination of those cases in which p -\- v is even and those in which it is odd. Denote the residues of least absolute value, mod p, of the products 1) by — a 1} — a 2 , •••, — a^ b 1} b 2 , •••, b\» and those of the products 2), mod q, by c 1? c 2 , • • *, c V y d 1) d 2) ' ' *, a p) 2 Monatsbericht der Berliner Akademie, December, 1872. QUADRATIC RESIDUES. I 37 the a's, fr's, c's and d's all being positive. Since p and q are dif- ferent from each other, one must be the greater. Assume q > p. We divide now the integers c lt c 2 , ',c v , all of which being resi- dues of least absolute value, mod q, belong to the system 1,2, into two classes according as they are greater or less than p/2. The system of those which are < p/2 we denote by C x and the system of those > p/2 by C 2 . Let v 1 denote the number of the integers C lf and v 2 that of the integers C 2 . The proof now falls naturally into the following four parts : i. That the integers, C lf are identical with the b's and therefore together with the a's make up the system p—i whence /* + v = - f- v 2 . ii. That the number, v 2 , of the integers C 2 is odd or even according as the number (p-{-q)/4 is or is not found among them. iii. That (p + q)/4 occurs among the integers C 2 , and hence v 2 is odd, when and only when we have simultaneously p = 4h — 1 and q = 4k + I. iv. That therefore p-f- v is odd when and only when simul- taneously p = 4h — 1 and q = 4k — 1 . The proof will be rendered more intelligible if we consider first the relation between the four parts into which we have divided it. Suppose that we have proved i, then (I) a) - (- ■>--. I38 QUADRATIC RESIDUES. and to prove our theorem it is sufficient to show that (p — i)/2 + v 2 is odd when and only when p = 4h — 1, q=^4k — I. It is evident, however, that since (p — 1)/2 is even or odd according as p = 4k + 1 or 4/t — 1, to show that (p — 1 )/2 + v 2 is odd when and only when p = 4h — 1, q = 4k — 1, it is suffi- cient to show that v 2 is odd when and only when p = 4h — 1, q = 4k-\-i. Now the number {p + q)/4 is less than q/2 and greater than p/2 and hence, */ an integer, is either one of the integers C 2 or one of the d's. But (p + q)/4 is an integer only when p = 4h + 1, q — 4k — 1 or p = 4h — 1, q = 4k-\-i, and hence can therefore evidently never be among the integers C 2 in the cases p =4/1+ i } q=^k -4- 1 ; and p=4h — 1, q = 4k — 1. If now we can show that (p -\-q)/4 always occurs among the integers C 2 when p=4h — 1, q=4k-\-i, and never when p=4h-\-i f q = 4k — 1, then to show that v 2 is odd when and only when p = 4h — 1, q = 4k-{-i, it will be sufficient to show that v 2 is odd when and only when (p + q)/4 occurs among the integers C 2 . Therefore to show that (p — i)/2 -\-v 2 is odd when and only when />=4/j — i, q = 4k — 1, it will be sufficient to show that (p + q)/4 occurs among the integers C 2 when and only when p = 4h — 1, p = 4k -f- 1. Our idea is there- fore to show that the three conditions p = 4h—i, q = 4k + i, v 2 odd, one of the integers C 2 , 4 are equivalent, whence it will follow that (p — i)/2 + v 2 is odd when and only when p = 4h — 1, q = 4k — 1. i. If any integer of the system p — i be not an a it must be a b ; for as we have already shown (Th. 4), the a's and b's together make up this system. The integers C x QUADRATIC RESIDUES. I 39 belong, however, also to this system, hence each of the integers C x must be either an a or a b. We shall show that each b is iden- tical with one of the integers C x ; also that no a is identical with any of the integers C x and hence the fr's and the integers C x coin- cide. Let bi be any one of the b's, and hiq that product of the system i) whose residue of least absolute value, mod p, is bi. P We have then hiq = bi, mod p ; o < hi < -; that is, hiq = kip-{-bi, 3) where ki is an integer such that P o< kip<hiq<-q, and hence o<*»<f. Therefore k ip is one of the products of the system 2). But from 3), we have kip = — bi, modg, where P o<bi<^. Hence bi is one of the integers C x . But bi is any one of the b's ; hence each b is identical with one of the integers C v Let now a } - be any one of the o's and hjq that product of the system 1) whose residue of least absolute value, mod p, is — ctj. We have then hjq = — dj, mod p ; that is, hjq = kjp — a ; -, 4) where kj is an integer > o and < q/2 ; for from 4) __hjq + aj kj ~~P ' P P and hence, since o < a y < -, and o < hj < -, — 2 140 QUADRATIC RESIDUES. we have 2*2 n ^ hi ^ O <v K) <. , that is, o<^<*+\ which gives, since k, and (g -f i)/2 are integers, Hence kjp is one of the products 2), and since from 4) it follows that kjp==aj, modg, dj is a d and therefore not one of the integers C t , But a, is any- one of the a's ; hence no a can be identical with one of the integers C v Hence the a's and the integers C x coincide, and therefore the a's and the integers C x make up the system /- I 1,2, , 2 Therefore fi -\- v = \- v 2 . ii. To prove now that the number, v 2 , of the integers C 2 is odd or even according as the number (p -j- q)/4 is or is not found among them, let Ci be one of the integers C 2 and kip = — a, modg. Here ki can not be (q — 1)/2, for we have 2 2 2 that is, / ss - £, mod a, 2 2 where (g — p)/2 is evidently positive and less than a/2, and hence one of the a"s. Therefore to each product, kip, of the system 2), whose residue of least absolute value, mod q, taken positively is an integer of C 2 , QUADRATIC RESIDUES. I4I there corresponds, since a product kjp, (fcj+ (q — i)/2), of the same system, such that ^ 2 *' We shall show now that the residue of least absolute value, mod q, of kjp, taken positively, is also one of the integers C 2 . Multiplying 5) by p, we have whence £•/ = ? + — — kj>, or k.p m tilt - kj>, mod q, and hence /£•/» ■ — — kj>, mod q. Moreover, since kip = — a, modg, we have k,p = — + c p mod q. ^ P Q But since - <*/<-, 2 * 2 we have £<£xf — *<'£ 22 2 Hence ^ is one of the integers C 2 . Putting /+_? - ^ = r., 6) we see that if kip, kjp, be two products of the system 2), such that ki and kj are connected by the relation 5), and if the residue of least absolute value of kip, mod q, be — d, where C\ is one of the 142 QUADRATIC RESIDUES. integers C 2 , then the residue of least absolute value of kjp, mod q, is — Cj, where Cj is also one of the integers C 2 . Hence to each integer d of C 2 there corresponds in this manner another integer Cj of C 2 and it is evident that unless it should happen that there is one (or any odd number) of these pairs whose integers are identical, the number, v 2 , of the integers C 2 will be even, but if the two integers composing each of any odd number of these pairs be identical, v 2 is odd. If a = Cj, then from 6) it follows that 4 * 4 Hence there is at most one pair whose integers are identical and this case will occur when and only when (p + q)/4 is one of the integers C 2 . Hence v 2 is odd or even according as (p-\-q)/4 does or does not occur among the integers C 2 . iii. To prove now that (p-\-q)/4 occurs among the integers C 2 , and hence v 2 is odd, when and only when we have simulta- neously p — ^h — i, q = 4k + i, we observe first that P P + 9 g 2 4 2 and hence, if (p-\-q)/4 be an integer, it is either one of the integers C 2 or a d. In order now that (p-\-q)/4 may be one of the integers C 2 it is necessary and sufficient that there shall be one, kp, of the products 2) such that / + Q kp m — , mod q ; that is, it is necessary and sufficient that there shall exist two integers h and k such that kp-kq- P -±Z, 7 ) QUADRATIC RESIDUES. 1 43 and k < - • 2 From 7) it follows that we must have ( 4 k + i)p=(4h—i)q, 8) and hence 4k -f- 1 divisible by q. q But we have k < - , 2 and hence 4& + 1 < 2g. Therefore g = 4k + 1, and consequently from 8) it follows that p = 4 h—i; that is, in order that the required integers h and k may exist, p and q must have these forms. Moreover, when p and q have these forms the required integers h and & evidently do exist. Hence p = 4h — 1, q = 4k -\- 1 i sa necessary and sufficient con- dition that (p + q)/4 shall be one of the integers C 2 . Therefore v 2 is odd when and only when we have simultaneously p = 4h — 1, and q = 4k + 1. iv. To prove now that /* + v is odd when and only when we have simultaneously p = 4h — 1, q = 4k — 1, we examine the equation j_ P — T 1 p-rv= — r-"2 and observe that p = 4h-\- 1, q = 4k + 1 gives even, v 2 even, fi-\-v even, 2 /> — 1 p = 4h -f- 1, g = 4£ — 1 gives even, v 2 even, ft + v even, p = 4h — 1, q = 4k-{-i gives odd, v 2 odd, /a -j- v even, 2 p = 4J1 — 1 f g = 4^ — 1 gives odd, v 2 even, fi-{-v odd. 144 QUADRATIC RESIDUES. Therefore (| ) (J ) = , when at least one of the positive primes p and q has the form 4W+I ,a„d (f)(J)— * when both have the form 4.11 — 1. 4 § 11. Determination of the Value of (a/p) by means of the Quadratic Reciprocity Law, a being any Given Integer and p a Prime. Before discussing the question of what odd prime moduli is a given positive odd prime a quadratic residue, which we shall be able to answer by means of the Quadratic Reciprocity Law, we shall illustrate upon an example how greatly the use of this law simplifies the determination of the value of (a/p), where a and p are both given integers and p an odd positive prime; that is, the determination whether the congruence „r 2 ==a, modp, has or has not roots. Let ^ = 365, mod 1847, be the congruence under discussion, 1847 being a prime. 1 We have V1847/ V1847/ V1847/ 847/ \i847/ \i847> Then since 5 is a prime of the form 4^+1, we have §47 N V1847/ v 847/ V 5 and since 1847 = 2, mod 5, 5 being of the form 8n — 3. Hence I — — \1847 1 Dirichlet-Dedekind : p. 103. QUADRATIC RESIDUES. 1 45 Likewise since 73 is of the form 411 + 1, and 1847 = 22, mod 73, we have (is^j = V73-) = (jj) = \y 3 ) xjj) ■ But (£)-,; since 73 is of the form 8n -\- 1, and therefore VT847/ = (73/ ' Again since 73 is of the form 4» + * an d 73 = 7, mod 11, (M)-(H)-(f,)- Since 7 and 11 are both of the form 471 — 1, (f.) =-(")- ©-©©—■ Therefore (^) = (-1) (_i) = i ; that is, 1 ) is solvable. Its roots can be shown to be ± 496. § 12. Determination of the Odd Prime Moduli of which a Given Positive Odd Prime is a Quadratic Residue. Let q be an odd positive prime. We are to determine for what positive odd prime values of p the value of (q/p) is 1, for what — 1. By means of the Quadratic Reciprocity Law we are able to make the solution of this problem depend on that of the simpler one, which we have already solved ; that is, the division of all rational integers into two classes, one of which contains all resi- dues of q and the other all non-residues. Let r x ,r 2 , '-,r t and n x ,n 2 , ••-,«* be respectively the incongruent quadratic residues and non-residues of q. Then an integer is a 10 I46 QUADRATIC RESIDUES. residue or non-residue of q according as it is contained in one of the forms r t + kq, r 2 -\-kq,-",rt-\-kq, 1 ) or in one of the forms n 1 + kq,n 2 + kq,--',n t + kq. 2) It is necessary now to distinguish two cases according as q has the form 411 + 1 or 4^ — 1. i. g =411+1. (*)-(& Then ft \q that is, q is a quadratic residue or non-residue of p according as p is a quadratic residue or non-residue of q. Hence q is a residue of all positive odd primes contained in the forms 1) and a non- residue of all positive odd primes contained in the forms 2). Ex. Let q = 13. The residues of 13 are 1, 3, 4, 9, 10 and 12, the non-residues 2, 5, 6, 7, 8 and 11. Hence 13 is a residue of all primes of the forms 1 +*I3&, 3 + I*3*j 4 + 13^ 9 + 13^ Jto + 13&, 12 + 13k, and a non-residue of all primes of the forms 2 + izk, 5 + izk, 6 + 136, 7 + 13^, 8 + 13k 11 + 13^ ii. q = 4.n — 1. We must further divide this case into two parts according as p has the form 4m + 1 or 4m — 1. a) p = 4tn-\-i. & - (£) Then pi \q and q is seen to be a quadratic residue of all primes of the form 4m + 1 contained in the forms 1 ) and a non-residue of all primes of the form 4m + 1 contained in the forms 2). b) p = 4m — 1. (JHf) • QUADRATIC RESIDUES. 1 47 and q is seen to be a quadratic residue of all positive primes of the form 4m — 1 contained in the forms 2) and a quadratic non- residue of all positive primes of the form 4m — 1 contained in the forms 1). The primes p are in this case seen to be subjected to two con- ditions, first that they shall give with respect to the modulus 4 the residues 1 or — 1, and secondly with respect to modulus q the residues r ly r 2 , --,r t or n 19 n 2 , • • • , n t . By Chap. Ill, § 14, we can find the forms which the numbers must have in order to satisfy both of these conditions. Ex. 1 Let q = 19. The residues of 19 are i, 4, 5, 6, 7, 9, 11, 16 and 17, and the non-residues 2, 3, 8, 10, 12, 13, 14, 15 and 18. Hence 19 is a residue of all positive primes of the form 4m -\- 1 con- tained in the forms 19* + i, 19* + 4, I9& + 5, 19^ + 6, 19^ + 7/ 19*4-9, 19* + n, 19* + 16, 19* 4" 17, 3) and of all positive primes of the form \m — 1 contained in the forms 19* 4" 2 > 19^ 4"3? I9# + 8, 19* 4~ IO > J 9^ 4~ I 2 , 19* + 13, 19* 4-14, 19&+ 15, I9& 4- 18. 4) On the other hand 19 is a non-residue of all positive primes of the form 4W — 1 contained in the forms 3) and of all positive primes of the form 4m -f- 1 contained in the forms 4). By Chap. Ill, §14, we may combine the two conditions thus imposed upon p into a single one and say that 19 is a quadratic residue of all primes of the forms 76*4-1, 3, 5, 9, 15, 17, 25, 27, 31, 45, 49, 51, 59, 61, 67, 71, 73, 75, and a quadratic non-residue of all primes of the forms, 76*4-7, 11, 13, 21, 23, 29, 33, 35, 37, 39, 41, 43, 47, 53, 55, 63, 65, 69. § 13. Determination of the Odd Prime Moduli of which any Given Integer is a Quadratic Residue. It was shown in § 10 that the solution of this problem could be made to depend upon the solution of the three simpler problems, to determine : 1 Wertheim : p. 220. I48 QUADRATIC RESIDUES. i. Of what odd prime moduli — 1 is a quadratic residue. ii. Of what odd prime moduli 2 is a quadratic residue. iii. Of what odd prime moduli another positive odd prime is a quadratic residue. These problems have all been solved and we are now prepared to solve the general question proposed originally in §2;- that is, to determine^ for what _o^_j^ime~^aiues of p the value ofJj^Q^, is i and for what — 1, a being any given integer. Assuming that . a contains no square factor and by pi denoting — 1 or any positive prime factor of a, we have for. each pi two systems of forms, one of which contains all positive odd primes of which pi is a residue, the other all positive odd primes of which pi is a non-residue. The positive odd primes of which a is a residue will be those which are contained in none or an even number of the second set of forms. Having obtained for each pi these two systems of forms the solution of the problem reduces to that of finding an integer which gives certain residues with respect to ^ach one of a series of moduli (Chap. Ill, §14). A single example must suffice here to illustrate the application of this method. For an extended discussion of it with numerous examples see Wertheim, pp. 221, and for the solution of this problem as well as the more general one, where the modulus is 'also composite, see Dirichlet- Dedekind, Bachmann and Mathews, where by an extension of Legendre's symbol a simplification is effected. Ex. Let a = — 15. (^)=(t)(I)0) Two cases must now be distinguished according as p has the form 4fei + 1 or 4k, + 3. If /, = 4*1 + 1, (t?)'** and (- ) = ( -- ) = 1 when /> = ^2-^1^ and as — 1 when p as 3&? -f 2 If /,^4^ + 3 , ^)=:_I, QUADRATIC RESIDUES. 1 49 and [j\ =- (0 = I when p = 3 k 2 + 2, and rs — 1 when p = $k 2 -f 1. In both cases ©-(3 1 when p = 5& 3 + X or 5& 3 + 4, and = — 1 when /> = 5& 3 + 2 or 5^3 + 3. In order now that — 15 shall be a residue of p, p must have such a form that either none or two of the symbols ( — i/p), (3/p), (5//O have the value — 1. Hence — 15 is a residue of all primes which are contained simulta- neously in the forms of one of the following sets : 4&i + i, 3&2 -f 1, 5^3 + 1, which give p = 60k + 1, 1 ) 4&i + i, 3&2 + 1, 5^3 + 4, which give p = 60k -f 49, 2) 4&1 + 1, 3& 2 + 2, 5^3 + 2, which give p = 60k + 17, 3,) 4^1+1, 3^2 + 2, 5& 3 + 3, which give /> = 60k + 53, 4.)- 4&i + 3, 3& + 1, 5^3 + 1, which give p = 60k -f- 31, 5) 4&i + 3, 3&2 + 1, 5^3 + 4, which give p = 60k + 19, 6) 4&i "k 3, 3k +.2, 5^3 + 2, which give /> = 6ofc + 47, 7) ' 4&i + 3, 3^2 + 2, 5^3 + 3, which give p = 60k + 23. 8)- ■ V We can easily combine 1) and 5), 8) and 6), 3) and 7), 4) and 8), and ♦obtain as the forms of the positive odd primes of which — 15 is a residue 30& + 1, 17, 19, 23. Similarly we find that — 15 is a non-residue of all positive primes contained in the forms 3o£ + 7, n, 13, 29. j. § 14. Other Applications of the Quadratic Reciprocity Law. We shall now give a few theorems in the proof of which the Quadratic Reciprocity Law and its two subsidiary theorems will be found useful. Theorem 6. There are an infinite number of positive primes of each of the forms 4n-\- i and 4n — i. 1 Observing that every prime is of one of these forms, we pro- 1 See Chap. II, § 6. I50 QUADRATIC RESIDUES. ceed to prove that there is an infinite number of primes of the form 4n -f- 1. Suppose that there is only a finite number of positive primes Pi> p2> ' ' '■■> Ps, of the form 4.W + 1. Form the integer (2p 1 p 2 --p 8 y + i=a, which is of the form 4% -j- 1. It is divisible by no prime q of the form 4% — 1, for, if this were the case, we should have {2p x p 2 --- p s y = — 1, modg; that is, — 1 would be a quadratic residue of q which is impossible because q is of the form 471 — 1. Moreover, a is not divisible by any of the primes 2, p lt p 2 , •••,/>«. Hence a is itself a prime of the form 4n-\- 1, different from each of the primes p lt p 2 , •••,^, or is a product of such primes. But this is contrary to our assumption that there are no primes of the form 4M+ 1 other than p lt p 2 , ••*,£«. Therefore the number' of positive primes of the form 471 — |— 1 is infinite. To prove now that there is an infinite number of positive primes of the form 4% — 1, we assume as before the contrary to be true; that is, that there are only a finite number of positive primes Qi> <?2> "'fit °f tne form 4% — 1, q t being the greatest. Form the integer zq 1 q 2 • • • q t + 1 = b. It is greater than q t and is not divisible by any of the primes 2 > Qi> #2> "'i9.t* Hence, if it be not prime, its prime factors must all be of the form 411 -{- 1. Let 2q 1 q 2 '--q t + i=p 1 p 2 '"pr 9 1) where p x == 1 /> 2 = i mod 4. prwmi ■ Multiplying these congruences together, we have • PiP 2 '"pr=i, mod 4, QUADRATIC RESIDUES. I 5 I whence 2^ x g 2 • • • qt + i = i, mod 4, and hence QiQ2'"Qt = 0, mod 2. 2) But 2) is impossible since q 1 ,q 2 ,'",qt are all primes of the form 4n — 1. Hence 1 ) is impossible and b is either itself a prime of the form 4n — 1 or is a product of primes of this form, all of which are greater than qt. Therefore the number of positive primes of the form 4-n — 1 is infinite. Theorem 7. Every prime of the form 2 2 + 1 has a primitive root. 3. ■ Let /> = 2 2 " +1. If 3 be a primitive root of p, then each of the (p — 2) powers of 3 must be incongruent to 1, mod p. tt) If, however, 3*=== 1, mod p, where o<t<p — 1, p being positive, then, by Chap. Ill, Th. 25, it follows that P — iso, mod/, ?>i.ji_ and, since p — 1 = 2 2 " , / — 2 W l — 4 , and the greatest possible value of t will be 2 2n _1 . In order, there- fore, that 3 may be a primitive root of p, it is necessary and suffi- cient that the following 2 n — 1 incongruences should hold 3 *i, 3 22 4u, }, mod p. 3 2 *ii A sufficient condition for this is that the last of these incon- gruences should hold, for if any one of the previous ones did not hold, all following ones would not hold. We have therefore only to prove ^~ l ^i, mod/>; I52 QUADRATIC RESIDUES. that is 3 2 ^1, mod/>. 3) But when 3) is satisfied, 3 is a quadratic non-residue of p, and vice versa. Hence we have only to prove (3//O = — I. Since p is of the form 4^+1, we have /" W 4) (!)-(!) But 2 = — I, mod 3, whence 2 2 " = ( — 1 ) 2 " ■■ 1 , mod 3. Therefore 2 2n + 1 = 2, mod 3, whence from 4) it follows that ©- Therefore 3 is a primitive root of every prime of the form 2 2 " + I. The theorem just proved bears an interesting relation to the problem of the construction of regular polygons of a prime num- ber of sides with ruler and compasses ; the construction is possible only when p is a prime of the form 2 2 " -f- 1, and can be accom- plished by means of a primitive root of p. 1 Theorem 8. Every positive prime p of the form 4q -f- 1, where q is a positive prime, has 2 as a primitive root. If 2 be a primitive root of p, then each of the p — 2 powers of 2 2,2 2 , ..-,2P- 2 must be incongruent to 1, mod p. If, however, 2 appertains to an exponent t, mod p, less than p — 1, then 2*e=i, mod/>, 5) 1 See Klein : Ausgewahlte Fragen der Elementar Geometrie, p. 13. Gauss: Disq. Arith., Sect. Sept. Works, Vol. I, p. 412. Bachmann : Die Lehre von der Kreisteilung, p. 57 and Vor. 7th. QUADRATIC RESIDUES. I 53 and by Chap. Ill, Th. 25, p — 1=0, mod t, whence 4q mm o, mod /. Hence, since q is a prime, we can have as possible values of t only 2, 4, q or 2q. It is necessary and sufficient to show that 2 4 =)si, mod/>, and 2 2 «=4=i, mod/>, for, if 2 2 a=i, mod p, then 2 4 a»l, mod/>, and, if 2«s=i, mod />, then 2 2 «e= i, mod/>. To prove 2 4 4a 1, mod /> ; that is, 15^0, mod/>, it is sufficient to notice that the only primes which divide 15 are 3 and 5, neither of which is of the form 4*7 -+- 1, when q is a prime. Hence 2 4 ^i, mod/>. To prove 2 2 «^ 1, mod /> ; that is, 2 ( *>- 1) / 2 4= 1, mod p, we need only show that (7)— we have I— J = (- 1) 8 = (- l)^*- - 1, for if ^ = 2, />, =4*7 -j- 1, is not a prime and therefore q is always odd, whence it is evident that 2q 2 + g is an uneven integer. Hence 2 2 «4 S l > mod/>. Therefore 5) holds for no value of t less than p — 1. Hence 2 is a primitive root of every positive prime of the form 4q -\- 1 when q is a positive prime. Examples. 1. Determine the prime moduli of which 30 is a quadratic residue. I 54 QUADRATIC RESIDUES. 2. Has the congruence jtr 2 = H35, mod 231 1, roots ? 3. Solve the congruences : 1 a) $x 2 — &r — 3 = 0, mod 23. x = 8 or 12, mod 23. b) 3*^ + 4* + 5 = 0, mod 20. .r = — 3, — 5, 7, 5, mod 20. c) Jx 2 — 3;tr-(- 11 =0, mod 19. ;fe=5, 9, mod 19. d) 5a- 2 — $x — 2 = 0, mod 12. x = — 2, 1,2, 5, mod 12. e) 3jf 2 + 4^+ 9 = 0, mod 12. arss — 3, 3, mod 12. /) Z x% + x — 4 — °> mod 10. x = 1,2, 6,7, mod 10. 4. Show that among the numbers of a reduced residue system, mod p n , where p is a prime different from 2, there are exactly as many quadratic residues as non-residues of p n . 2 5. Show that every quadratic residue of p is also a quadratic residue of p n , and that every non-residue of p is also a non- residue of p n . s 6. The numbers a and p — a, where p is a prime, have the same or opposite quadratic characters, mod p, according as p is of the form 4W -}- 1 or 4n — 1 . 1 Wertheim : Anfangsgriinde der Zahlenlehre, 1902, pp. 320-322. This book contains many numerical examples and should be consulted by every one interested in such work. It also contains many interesting historical notes and some useful tables, and is in many ways a good book for a beginner to read. 2 Gauss : Disq. Arith., Art. 100 ; Works, Vol. I. 3 Ibid., Art. 101. CHAPTER V. The Realm k(i). 1 § i. Numbers of k(i). Conjugate and Norm of a Number. The number V — I, that we shall as usual denote by i, is defined by the equation x- + I = o I ) which it satisfies. Every number of k(i) is a rational function of i with rational coefficients (Chap. I, §3), and since by means of the relation i 2 = — 1 the degree of any rational function of i may be reduced so as to be not higher than the first, every number, a, of k(i) has the form a = a 1 + b 1 i a 2 -f b 2 i ' where a lt b ly a 2 , b 2 are rational numbers, or, multiplying the numer- ator and denominator of this fraction by a 2 — b 2 i, we have a x a 2 -f- b x b 2 a 2 b x — aj> 2 . <*2 ■+- b 2 a i + b, 2 that is, every number, a, of k(i) has the form a = a -{- hi, where a and b are rational numbers.'' The other root — i of the equation 1) defines the realm k{ — i) conjugate to k{%) (Chap. I, § 4). These two realms are identical, 1 Gauss : Th. Res. Biquad. Com. Sec, Works, Vol. 2, p. 95, f. f. Dirichlet- Dedekind : § 139. Weber : Algebra, Vol. I, § 173. Dedekind : Sur la theorie des nombres entiers algebrfuques ; Bulletin des Sc. Math., 1st Ser., Vol. XI, and 2d Ser., Vol. I. Bachmann : Die Lehre von der Kreisteilung, 12th Vor. Cahen: pp. 354-367. 2 Throughout the remainder of this book letters of the Latin alphabet will always denote rational numbers (except in £(0, where * = V — 1) while letters of the Greek alphabet will denote the general numbers of the realm under discussion, which may or may not be rational numbers. 155 I56 THE REALM k(i). • for i is a number of k( — i) and — i is a number of k{i) (Chap. I, §3). The number a — bi, obtained by putting — i for i in any number a, =a -f- bi, of k(i), is the conjugate of a and is denoted by a'; for example, 3 + 2.1 and 3 — 2,i are conjugate numbers (Chap. I, §4). A rational number considered as a number of k(i) is evidently its own conjugate. It is easily seen that the conjugate of a product of two or more numbers of k(i) is equal to the product of the conjugates of its factors; that is, if fx = ap, then /i=a , p'. The product of any number, a, of k(i) by its conjugate is called the norm of a and is denoted by n[a] ; that is, n[a + bi] = (a + bi) (a—bi) =a 2 + b 2 For example: *(3 + 2i]=(s + 2i) (3 — 21) = 13, and n[s]==5'5 = 25- We observe that the norms of all numbers of k(i) are positive rational numbers. Theorem i. The norm of a product is equal to the product of the norms of its factors; that is, n[a/3] =n [a] -n[/3]. For n[ap]=ap-a'F = « [a] •«[/?]. Every number, a, of k(i) satisfies a rational equation whose degree is the same as that of the realm, that is, the second, and whose remaining root is the conjugate of a, for the equation having for its roots a,=a-\- bi, and a', =a — bi, where a and b are rational numbers, is x 2 — 2ax + a 2 -\-b 2 = o; 2) and this is of the form x 2 + px + q = o, 3) where p and q are rational numbers. THE REALM k(i). I 57 If b=o, that is, if a = a', the equation 2) is reducible, becoming O — a) 2 = o, and the rational equation of lowest degree that a satisfies is x — a = o If ,&=j=o, that is, if a =4= a', the equation 2) is irreducible, and hence is the single rational equation of lowest degree and of the form 3) satisfied by a (Chap. I, § 2). We observe, therefore, that the numbers of k(i) fall into two classes according as the irreducible equations of lowest degree satisfied by them are of the first or second degree. Those of the second class, that is, those which satisfy irreducible rational equa- tions of the same degree as that of the realm, are called primitive numbers of k(i). The numbers of the first class, that is, those which satisfy irre- ducible rational equations of a degree lower than that of the realm, are called imprimitive numbers of k(i). The imprimitive numbers of k(i) are evidently the rational numbers. All numbers of the realm R being included among those of the realm k(i), R is said to be a sub-realm of k(i). It is easily seen that k(i) may be defined by any one of its primitive numbers, but by none of its imprimitive numbers. The constant term of the rational equation of the form 3) whose roots are a and a' is seen to be n[a]. In general, each number a, of a realm, k(&), of the nth. degree satisfies a rational equation whose degree is the same as that of the realm and whose remaining roots are the n — 1 conjugates of a (see Chap. VIII, Th. 4). §2. Integers of k(i). To ascertain what numbers of k(i) are algebraic integers we may consider separately the two classes of numbers of the realm, the imprimitive numbers being at once disposed of by remember- ing that a rational number is an algebraic integer when and only when it is a rational integer. I58 THE REALM k(i). To determine when a primitive number a is an algebraic integer, we observe that the necessary and sufficient condition that a shall be an algebraic integer is that the coefficients of the single rational equation of lowest degree, x 2 + px + q = o, satisfied by a shall be integers (Chap. II, Th. 4). But — p = a-\-a', and q = aa' and hence the necessary and sufficient conditions that a shall be an algebraic integer are that a + a! and aa' shall be rational integers. 2 If we write a in the form a -\- bi, where a = a x /c x , and b = b x /c x , #!, b x , c x being rational integers with no common factor, these conditions become - J — - — L -f — = — r= a rational integer, 1) c 1 c x c ± ( °±±M ) ( ^M ) . k^l = a rational int e g e r . 2 ) One at least of the three following cases must occur: i. q4=2 or 1; ii. c 1 = 2; iii. ^=1. We shall show that i and ii are impossible. i. If c 1 =%=2 or 1, then by virtue of 1) o x and c 1 would have a common factor that by virtue of 2) would be contained in b x also. But this is contrary to our hypothesis that a lt b lt c x have no com- mon factor. Hence i is impossible. ii. If c x = 2, then by virtue of 2) a x 2 + b x 2 would be divisible by 2 2 and hence a x and b x each by 2 ; that is, a x , b x , c x would have the common factor 2, which is contrary to our hypothesis. Hence ii is impossible. Hence c x = 1 ; that is, a and b are rational integers. 2 This is a special case of the general theorem that a necessary and sufficient condition for an algebraic number a to be an integer is that all the elementary symmetric functions of a and its conjugates shall be rational integers. THE REALM k(i). I 59 Thus all integers 1 of k(i) have the form a + bi, where a and b are rational integers, and all numbers of this form are integers of k(i). If b = o, we obtain the rational integers. The conjugate of any integer of k(i) is evidently also an integer, and the norm of any integer of k(i) is a positive rational integer. We observe that in k(i), as in R, the sum, difference and product of any two integers are integers. 2 §3. Basis of k(i). Any two integers a> lf o> 2 of k(i) are said to form a basis of the realm if every integer of the realm can be represented in the form a 1 o) 1 + a 2 w 2 , where a lt a 2 are rational integers. 3 It is evident that all numbers of the form a^ -f- a 2 w 2 are in- tegers of k(i). We have already seen that I and i form a basis of k(i) ; that they are not the only integers of k{i) having this property is easily shown. For example: 1 + *, 3 + 2/ is also a basis; for if a-\-bi be any integer of k(i), then from a + bi = (h(i -ft) +02(3 + 20, we have ai + 3a* = a, fli + 2a 2 = b, giving ai = — 2.a + 3b, Oz = a — b, which are rational integers since a and b are rational integers. We have a + &*•=(- 2a + 3 fc) (1 +0 + (<*-&) (3 + 2*')- 1 Throughout the discussion of k(i) the term integer wiH be used to denote any integer of the realm either complex or rational. 2 It is true, in general, that the sum, difference, and product of any two algebraic integers is an algebraic integer (see chap. IX, Th. 8, Cor. 2). 3 There exist in every realm of the nth degree n integers «i, w 2 , •••, <»n, such that every integer of the realm has the form 6 = a x Wi + a&z + • • • + (hio>n, where a u a 2 , •••, a n are rational integers. In the definition here given I have followed Hilbert (see H. B., §4). The basis defined above is some- times called a minimal basis of the realm (see Weber: Algebra, Vol. II, §H5). i6o THE REALM k(i). For example ; 8 + 5* = — ( i + 1) + 3 (3 + 2*) . Every integer of the realm is therefore expressible in the form ai(i+0 + *(3+*0» where at and 02 are rational integers. Hence 1 + i, 2>-\-2i is a basis. We observe that the determinant of the coefficients 1 of 1 + * and 3 + 2% is 1 1 3 2 this being a particular case of the following theorem. Theorem 2. // m xt o> 2 fo a basis of k(i), the necessary and sufficient condition that 1) where a lt a 2 , b lt b 2 are rational integers, shall be also a basis of k(i) is a t a. b x b 2 ± 1. 2) This condition is necessary; for, if c^*, o> 2 * be a basis, we have , 2 = & 1 *» 1 * + & 2 *o s 3) 4) 5) where fl^*, a 2 *, & x *, b 2 * are rational integers, and substituting the values of c^*, w 2 * from 1) in 3), we have g>i= (fl^*^ -f- 2 *^i) w i + ( a i*°% + a 2 *b 2 )<»2> From 4) and 5) it follows that a i*#i + #2*^1 = J > #i* fl 2 + G-2*&2 = °> ^i*«i + &2*&i = o, K*<h + b ** b 2 = 1, whence o x * a 2 * b* b* h b 2 ^1*^ + a 2*^2 ^1*^2 + ^2*^2 I O ass I. O I 1 We call a, b the coefficients of the number a«i + bw 2 , where *t, w 2 is a basis. THE REALM k(l). 161 Therefore = ± i. = ± i, The condition is also sufficient; for, solving i) for to x and <o 2 , we have, if 2) be satisfied, co 1 = ± (^2 w i* — ^2 W 2*)> and hence, if w, = c^ + C 2 W 2> be any integer of the realm, o>=± (cj> a + c 2 b x )<»* qz (c x a 2 + c&)*f ; that is, w = d x io x * + ^2 w 2*j where J x and d 2 are rational integers. Since there is an infinite number of different sets of rational integers a x , a 2 , b x , b 2 which satisfy the relation a x a 2 K b 2 there is an infinite number of bases of k(i). §4. Discriminant of &(f). The squared determinant formed from any basis numbers and their conjugates is called the discriminant of the realm, and is denoted by d. That d is the same, no matter what basis is taken, is evident from the last paragraph. For if w x , <o 2 and w^*, = a x <o x -j- a 2 w 2 , <o 2 *, = b x o) x -\- b 2 o> 2 , be any two bases, then I w i* w i «>!* ft). a 1 <o 1 + a 2 o) 2 , b x <a x -\- b 2 <o 2 2 a x <o x -f- a 2 (a 2 , b x <D x -\- b 2 <o 2 a x a 2 2 1 <a x w 2 2 w l o> 2 K b 2 t t I 0) x w 2 «l' co 2 Hence, since 1, i is a basis of k(i) d = = — 4- 11 1 62 THE REALM k(i). It that 32 THE REALM k(l). It is easily seen that if w^wg be any two integers of k(i) such tat 2 = d, then <*>!, w 2 is a basis of k(i). For example : i + * 3 + 2* I 2 — 4: i— i 3 — 2i \ Hence 1 + *, 3 + 2 * is a basis of &(0 as we have already seen. §5. Divisibility of Integers of k(i). Any integer, a, is said to be divisible by an integer, (3, when there exists an integer, y, such that a = Py. We say that /? and y are divisors or factors of a, and that a is a multiple of /? and y. Ex. i. We see that 8 + i is divisible by 3 + 21, since 8 + /= (3 + 20(2-0. Ex. 2. On the other hand 5 + 2i is not divisible by 1 + 3*', for there exists no integer of k{i) which multiplied by 1 + 3* gives' 5 + 2*. This can be shown as follows: If we set s + 2»=s (l4-30(*Hhy0i 1) we obtain jr=r|^, 37 = — |f ; that is, there are no integral values of x and y for which 1) will hold. Hence 5 + 3* is not divisible by 1 + 31. This can also be shown as follows: 5 + 2* (5 + 2p(i— 30 = n_ ii v i+3*' (i + 30d-30 " 10 As immediate consequences of the above definition we have the following : i. If a be a multiple of (3 and ft be a multiple of y, a is a mul- tiple of y, or more generally ii. // each integer of a series a,(3,y,8,---,bea multiple of the one next following, each integer is a multiple of all that follow it. THE REALM k(i). 1 63 iii. // two integers, a and /?, be multiples of y, then a£ + fa is a multiple of y, where $ and t] are any integers of the realm. It will be observed that iii depends not only upon the above definition but upon the fact that the sum, difference and product of any two integers of k(i) is an integer of k(i). If a be divis- ible by /?, then a' is divisible by (3' ; for, if a = £y, then a' = 0'y'. In particular, if a rational integer be divisible by any integer of k(i), it is divisible by its conjugate. Theorem 3. If a be divisible by /?, n[a] is divisible by n[(3]. For, if a = fiy, it follows from Th. 1 that n[a]=n[p]n[y], and hence that n[a] is divisible by n[(3]. The converse of this theorem is not in general true, as may be seen from the following example: If a = 8-f-*' and £ = 3 — 2t, n[a], = 65, is divisible by n[(3], = 13, but a is not divisible by /?; for putting 8+t= (3— *0(*+'j9» we obtain fractional values for x and y. The determination of the conditions under which n[a] divisible by n[fi] is a sufficient as well as necessary condition for a to be divisible by ft must be postponed until the unique factoriza- tion theorem has been proved for the integers of k(i). If two or more integers, a, /?, y, • • • , of k (i) be each divisible by an integer fx of k(i), fx is said to be a common divisor of a,P,y, •••- §6. Units of k(i). Associated Integers. We have seen that in the rational realm there are certain in- tegers, zh 1, called units, which are divisors of every integer of the realm. Evidently ± 1 have this property in k(i), and are therefore called units of k(J). We ask now whether there are any other integers of k(i) which enjoy this property. If there be such integers they must be divisors of 1, and conversely every divisor of 1 is a unit. Let e, = x -\-yi, be a unit of k(i) ; then ae=i, 1) 164 THE REALM k(i). where a is an integer of k{i). It follows that w[a]w[e] = 1, and hence w[c] = 1 ; that is, x* + y 2 =i. 2) That n[e] =1 is not only a necessary but also a sufficient con- dition that e shall be a unit, is evident from the fact that from it follows ee = 1 , and hence that € is a divisor of 1. From 2) .it follows that x=± 1, ;y = o; x — o, y=:±i, and hence e = 1, — 1,1 or — i, Therefore I, — i,i, — i are the units of k(i). That all these in- tegers are units of k{i) may easily be verified, since, if a -j- bi be any integer of k(i), we have a-|- bi=i(a-\-bi) = — i( — a — bi) =*( — ai-\- b) = — i(ai — b) Starting with the original definition of a unit as an integer which is a divisor of every integer of the realm, we obtain there- fore the three following equivalent definitions for the units of k(i): i. They are the divisors of 1. ii. They are those integers whose reciprocals are integers. Hence the reciprocal of a unit is a unit. iii. They are those integers whose norms are 1. Hence the conjugate of a unit is a unit. Two integers, a and (3, with no common divisor other than the units are said to be prime to each other. It is customary also to say that two integers, whose common divisors are units, have no common divisor. A system of in- THE REALM k(t). 165 tegers, a x , a 2 , •••,a„, such that no two of them have a common divisor other than the units are said to be prime each to each. As in the rational realm, two integers, m and — m, that differ only by a unit factor, are said to be associated, so in k(i) the four integers, a, — a, ia and — ia, obtained by multiplying any integer, a, by the four units in turn, are called associated integers. For example, the four integers 3 + 21', — 3 — 2i, — 2 -f- 3*, 2 — 3J are associated. We say also that a, — a, ia, — ia are the asso- ciates of a. Any integer that is divisible by a is also divisible by — a, ia and — ia. Hence in all questions of divisibility associated integers are considered as identical. It will be understood from now on that when two factors, a, (3, of an integer of k(i) are said to be the same, they are merely associated; that is, a = e/?, where c is a suitable unit. They may or may not be equal, equality .being understood in the ordinary sense ; that is, a 1 + b x i = a 2 + b 2 i, when and only when a x = a,, and b t = b 2 . If each of two integers be divisible by the other, they are asso- ciated, for let a/ft = y, then p/a=i/y. If now both y and 1/7 be integers, then y is a unit and a and (3 are associated. § 7. Prime Numbers of k(i). An integer of k(i), that is nut a unit and that has no divisors other than its associates and the units, is called a prime number ofk(i). An integer of k(i) with divisors other than its associates and the units is called a composite number. It will be observed that these definitions are identical with the corresponding ones in the rational realm. To ascertain whether any integer a, not a unit, is a composite or prime number, we have only to determine whether or not a can be resolved into two factors neither of which is a unit. We put therefore a = (a -\- bi) (c -\- di) and determine for what sets of integral values of a, b, c and d this equation is sat- isfied. If any one of these sets of values be such that neither a+ bi nor c + di is a unit, a is a composite number; but, if for every set of values one of these factors be a unit, a is a prime. 1 66 THE REALM k(i). Ex. I. To determine whether 3 is a prime or composite number of k(i). Put 3=(a + bi) (c + di) ; then 9 = (c? + b 2 )(c 2 + d 2 ), whence we have either 2 . F 1 ) or H C 2 +rf 2 = 3 J C 2 +rf 2 = 9 J Remembering that a, b, c and d must be rational integers, we see that 1) is impossible, while from 2) a-\-bi is a unit. Therefore 3 is a prime number of k(i). Ex. 2. To determine whether 7 + 41" is a prime or composite number of k(i). Put 7 + 4t'=(a + bi) (c + dt) ; then 65= (a 2 + b 2 )(c 2 + d 2 ), whence we have either a 2 + b 2 =5 l a 2 +fc 2 =i '2) Li) or I c 2 + <f=i 3 j <* + <** = 65 j low that a + bi is a unit, but = ± I, 1 fl=± I, &=±2, ] = ± 2, ) C=±2, rf = ± 3, J From 2) it would follow that a + bi is a unit, but 1) gives a=± 2, b = ± 1, ) a=± 1, b c = ± 3, d whence o + &*" = ± (2 + *) or ± (1 — 2/), 3) or a + bi = ± (2 — t) or ± ( 1 -f- 21) , 4) and c + dt = ± (3 + 2/) or ±(2 — 3?), 5) or c -j- di = ± (3 — 2O or ± (2 + 30 > 6) the four integers after each sign of equality being associated. It will be observed that this process gives us not only the divisors of 7 + 4i and its associates, but also the divisors of every other integer whose norm is 65; that is, of 7 — 4*', 8 + t, 8 — *, and their associates. Each one of the eight values of a + bi multiplied by any one of the eight values of c -J- di gives an integer whose norm is 65, and these sixty- four integers fall into four classes of sixteen each according to the one of the integers 7 + Ah 7 — 4h 8 + i, 8 — i, with which they are as- sociated. Each associate of each one of these four integers will be repeated exactly four times. Selecting by trial the divisors of 7 -f- 4*, we see that any integer from 4), multiplied by a suitable one from 6), gives 7 + 4*. Thus 7 + 4*'= (2 — 0(2 + 30- 7) Hence 7 + 4/ is a composite number. THE REALM k(l). 1 67 We have also, 7 + 41 = (— 2 + •) (— 2 — 3*') , = ( i + 2*')( 3 — 2»), = (— I" 2*)(— 3 + 2*), but these factorizations are looked upon as in no way different from 7) since the corresponding factors are associated. Hence 7 + 4* can be factored in only one way into two factors, neither of which is a unit. If now we attempt to factor 2 — i and 2 + 3*, we find that they are prime numbers, and hence we say that 7 + 4* has been resolved into its prime factors. Ex. 3. Resolution of — 23 + 41*' into prime factors. If we endeavor to resolve — 23 + 411 into two factors neither of which is a unit, we find that it can be done in seven different ways; that is, — 23 + 41*= (1+3*) ( 10 +nt), "" = (i + 5*)( 7+ 60, = (3 + 5*)( 4+ 70, = (l+ *')( 9 + 320, \ 8) = (2+ *')(— 1 + 21O, = (3 + 2*')( 1 + 13O, = (4+ 0(— 3 + nO- _ We find, however, that in each case either one or both of the factors is composite and we resolve the composite ones into the following factors all of which can easily be proved to be prime: i-f 3»= (i + 0(2 + 0; i+Sf= (i + 0(3+2*'); 3 + 5*"= (i + 0(4 + J 10+11*' = (3 + 2*')(4 + ; 7 + 6*= (2 + 0(4 + 0; 4 + 7* = (2 + 0(3 + 2*'). when these values are substituted in 8) we have in all seven cases — 23 + 41* = ( 1 + (2 + (3 + ») (4 + ; that is, if — 23 + 41 1 be resolved into factors all of which are prime, the resolution can be affected in only one way. It is now evident that we can, as in the case of the rational integers, represent every integer of k(i) as a product of its prime factors, and the last example renders it probable that the repre- sentation will be unique. We shall proceed to prove three theorems which will enable us to show that the integers of k(i) have indeed this all-important property. § 8. Unique Factorization Theorem for k(i). Theorem A. // a be any integer of k(i), and (3 any integer of 1 68 THE REALM k(i). k(i) different from o, there exists an integer fi of k(i) such that n[a — fjL/3]<n[p]. Let a/(3 = a+bi, where a = r-\-r lf b = s -\- s 1} r and ^ being the rational integers nearest to a and b respectively, and hence We shall show that /*, =r-{-si, will fulfill the required con- ditions. Since a/p — /* = r t + *i*, whence « [a//3 — fi] < I ; or, multiplying by «•[£], n[a — lip] <n[p]. Ex. If a 3= 5 + 2i, and = i -f- 3/, then a — 5 + 2 *' _ 1 1 _ 1 3 i and /* = 1 — i, therefore a — /*)3 = 5 -j- 2* — ( 1 — i) ( 1 + 3O = 1, and w[i] <ra[i -J- 3*]. The method given above for selecting /x evidently determines it uniquely unless either one or both of the quantities |^i|, |^i| be J, in which cases there are respectively 2 or 4 integers which satisfy equally the method of selection. There are, however, values of /x that satisfy the requirements of the theorem other than the one selected as above. In the ex- ample given above it would serve as well to take H = 2 — i or 1 — 21 ; for 5 _L. 2 ;_( 2 _;)( I _|_ 3 f)= = _3*; and n[ — p] <fl[l+3*] \ likewise 5 + 2* — (.1 — 2*) ( 1 + 3*) = — 2 + i, and n [ — 2 + *'] < n [ 1 + 3*] . THE REALM k(i). 169 It can be easily shown that there are in general (including the one selected as in the proof) two, three or four values of fi which satisfy the requirements of the theorem. The particular value of fi selected as above may be called the nearest integer to a/ft. The other possible values of n are found among the integers r t + s a i such that r 2 ,s 2 differ respectively from r l ,s l by 1. This will be made clearer by a graphical proof of the theorem to which we are led by its statement in the following form : // a/p be any number of k(i), there exists an integer /x of k(i) such that n[a//3 — fi] < I. Y -2+2i -l+2i 2i l+2i 2+2i -2+i -1 + i i 1 + i 2+i ■■»■ -2 -1 1 2 /^ P, Po -2-i -1-i -i 1 1-i 2-i / -2-2i -l-2£ -2i \l-2i p y 2-2i Representing as is usual the number x -\- yi by a point whose coordinates referred to rectangular axes are x and y, we see that the integers of k(i) are the points of intersection of a lattice 170 THE REALM k{%). formed by two systems of straight lines parallel respectively to the axes of x and y, and at the distance 1 apart. 1 Our problem is, given any number y of k(i), we are required to find all integers, ft, of k(i) such that »[y— rf<i- 1) Let G and N be points representing the numbers y, = a -\- hi, and v, =c -f- df, respectively; then every number, v, of k(t) such that ^.[y — v] < 1 is represented by a point lying within the circle of radius 1 de- scribed about G as a center, and conversely every number, v, of the realm represented by a point lying within this circle satisfies 1) ; for ( x — a y+ (y — b) 2 =i is the equation of a circle of radius 1 with center at G, and we have (c-ay+(d-by<i; that is n[y — v] < 1 when and only when the point (c,d) lies within this circle. The graphical solution of our prpblem consists therefore merely in describing a circle of radius 1 around the point representing y and observing what lattice points fall within it. In the figure the point G represents the number y = |£ — ^§* (see example above), and a circle of radius 1 described around G as a center is seen to enclose the three points P x , P 2 , P 3 , repre- senting the integers 1 — i, 2 — i, 1 — 2L Moreover, no other in- teger point falls within this circle. The integers 1 — i, 2 — i, 1 — 2» are all the values of fi which satisfy the condition n[y — /a] < 1, the integer 1 — i, which is the one given by the method of selec- tion used in the proof, being represented by the lattice point near- est to G. It is evident that the only possible values of fi are those repre- sented by the vertices of the lattice square in which the point G, representing y, lies. 'Cahen: p. 357. THE REALM k(i). I7I We see that two, three or four of these vertices will satisfy the required condition according as G lies in the unshaded, lightly shaded or heavily shaded portions of the square, the square being thus partitioned by describing from each vertex as a center an arc of a circle of radius i. G lt G and G 2 illustrate respectively the first, second and third cases. G ± and G 2 illustrate also the cases in which there are re- spectively two or four equally near lattice points (original method of selection is not unique). Returning once more to the theorem in its original form, we observe that it is equivalent to saying that for every integer /?, different from o, considered as a modulus there exists a complete residue system such that the norms of all the integers composing this system are less than n[/3]. This interpreted graphically implies that if we describe around the origin a circle with radius equal to "\/n>[p], that is, passing through the point representing /?, there will be among the integers represented by the lattice points lying inside this circle a complete residue system, modulus p. Theorem A is equivalent to saying that we can divide a by (3 so as to obtain a remainder whose norm is less than n[(3], the quotient being fi. In this form its analogy with Theorem A in R is even more clearly brought out. It enables us to do for k{%) exactly what we did in R by means of Theorem A ; that is, by an algorithm strictly analogous to that used in R to find a common divisor, 8, of any two integers a and /?, such that every common divisor of a and (3 divides 8. In other words, it enables us to prove that any two integers of k(i) have a greatest common divisor and to find it. 1 For example; let the two integers be 112 -f- * and — 57 + 79?'. We have II2 + * . = "^5 -8905/ whence „ = _ , _ ,- — 57 + 79* 9490 and 1 12 + i — (— 1 — i) (— 57 -f- 79J) = — 24 + 23?'. x See Dirichlet-Dedekind : p. 439. 172 THE REALM k{l). Likewise 57 ~r 9 l __ 3 1 5 5 5* > whence ih. = 3 — h — 24 + 231 1 105 and —57+ 79* — (3 — (— 24 -\-23i) = — 8 — 14*. Likewise - *4 + 23* = - 130 - 5*» f whence fJ ^ = _ 1 _ 2if — 8 — 14* 260 and — 24 -f 23* — ( — 1 — 2») ( — 8 — 14*) as — 4 — 71. Finally ~~ ~ I4 f — 2, whence ft = *M f* , — 4 — 71 and — 8 — V4» — (2) ( — 4 — 7*) = o. Therefore — 4 — 71 is the greatest common divisor of 112 + / and — 57 + 79*- Instead, however, of proving the existence of a greatest common divisor of any two integers of k(i), we shall proceed as in R, and shall prove the following theorem of which the greatest com- mon divisor theorem is an immediate consequence. Theorem B. If a and p be any two integers of k(i) prime to each other, there exist two integers, £ and 77, of k(i) such that a£ + p v =i. If either a or /? be a unit, the existence of the required integers, i, 7), is evident. We shall now show that, if neither a nor p be a unit, the determination of £ and rj can be made to depend upon the determination of a corresponding pair of integers £ lf i/ x , for a pair of integers, a lf p lf prime to each other and such that the norm of one of them is less than both n[a] and n[p]. Assume n[p] ^n[a], which evidently does not limit the gen- erality of the proof. By Th. A there exists an integer /* such that n[a — iip] <n[p]. Then p and a — /*/? are a pair of integers, a lt p i3 prime to each other and n[a — fxp] is less than both n[a] and n[p~\. If, now, two integers, £ lf rf 19 exist such that that. is, £& + (a — iiPh x =i, THE REALM k(i). 1 73 we have 0% + p(i 1 — ^J = i, and hence £ = , >?i> 7 7 == £i — Mi- The determination of &, ^ for a t , ^ may, if neither a x nor /^ be a unit, be made to depend similarly upon that df £ 2 > V2 f° r a pair of integers a 2 , fi 2 prime to each other and such that the norm of one of them is less than both n[a ± ] and w-[/y. By a continuation of this process, we are able always to make the determination of | and rj depend eventually upon that of £ n , rj n for a pair of integers a n , /?«, one of which is a unit. Since the existence of | M and rj n is evident, the existence of | and r] is proved. , We shall see later that, although the proof here given of the unique factorization theorem depends upon Th. A, there are realms in which the unique factorization theorem holds but Th. A does not hold. However, we shall see also that each of the three theorems B, C and the unique factorization theorem is necessary and sufficient for the validity of the other two. Cor. 1. If a and ft be any two integers of k(i), there exists a common divisor, 8, of a and ft such that every common divisor of a and ft divides 8, a>nd there exist two integers, $ and rj, of k(i) such that ai-{- firj = 8. The proof is the same as in R. We call 8 the greatest common divisor of a and /?. Cor. 2. // a lt a 2 , •••,a„ be any n integers of k(i), there exists a common divisor, 8, of a lt a 2 , • • • , a n such that every common divisor of a t , a 2 , • • • , a n divides B, and there exist n integers £i»£»> ••*,lii such that a£ t + a 2 i 2 H \- a n $n = 8. Theorem C. // the product of two integers, a and (3, of k{i) be divisible by a prime number, *, at least one of the integers is divisible by ir. Let a/3 = y7T, where y is an integer of k(i), and assume a not 174 THE REALM k(i). to be divisible by ar. Then a and tt are prime to each other and there exist two integers, £ and rj, of k(i) such that a£ + 7n?=i. 2) Multiplying 2) by p, we have and therefore 7r(y| + /fy) = /3, where y| + ft 1S an integer of &(*) ; hence (3 is divisible by tt. Cor. 1. // the product of any number of integers of k(i) be divisible by a prime number, ir, at least one of the integers is divis- ible by ir. Cor. 2. If neither of tzuo integers be divisible by a prime num- ber, tt, their product is not divisible by v. Cor. 3. // the product of two integers, a and /?, be divisible by an integer, y, and neither a nor (3 be divisible by y, then y is a composite number. Theorem 4. Every integer of k(i) can be represented in one and only one way as the product of prime numbers. Let a be an integer of k(i). If a be not itself a prime number, we have a = py, 3) where ft and y are integers of k(i) neither of which is a unit. From 3) it follows that n[a] = n[fi]n[y] , whence, since n[p] =j= 1 and n[y] =)=i, we have n[fi] and n[y] < n[a]. liftbe not a prime number, we have as before P— An, where p x and y x are integers neither of which is a unit, and hence 7t[/?J and n[y ± ] < n[p]. If p x be not a prime number, we pro- ceed in the same manner, and, since n[p], *[&], n[P 2 ], '" form a decreasing series of positive rational integers, we must after a finite number of such factorizations reach in the series p, p it p 2 , • • • a prime number w v Thus a has the prime factor tt x , and we have a = 7r x a x . THE REALM k(t). 1 75 Proceeding similarly with a x , in case it be not a prime number, we obtain a 1 = 7r 2 a 2 , where tt 2 is a prime number, and hence OL = tt x tt 2 OL 2 . Continuing this process we must reach in the series a, a lf a 2 , • • • a prime number tt„, since n[a], n[a x ], n[a 2 ], •• • form a decreas- ing series of positive rational integers. We have thus CL TT x TTc,TT% ' ' ' 7Tfi) where the tt's are all prime numbers ; that is, a can be represented as a product of a finite number of factors all of which are prime numbers. It remains to be proved that this representation is unique. Suppose that a = p x p 2 p 3 • • • pm is a second representation of a as a product of prime factors. It follows by Th. C, Cor. i from ir i" , 2 ir 3 * ' * **■ ~ P1P2PZ ' ' ' pm, 4) that at least one of the p's, say p lf is divisible by v u and hence associated with 7r ± ; that is, p 1 = e 1 7r 1 , where c x is a unit. Dividing 4) by tt x , we have tt 2 tt 3 • • • ir n = e x p 2 p 3 ■ ■ • p m . From this it follows that at least one of the remaining p's, say p 2 , is divisible by ir 2 , and hence associated with it. Thus p 2 = e 2 Tr 2 , where c 2 is a unit, and hence 7T 3 • • • 7T n == Z\£oP3 ' ' ' pm- Proceeding in this manner, we see that with each n there is associated at least one p, and, if two or more tt's be associated with one another, at least as many p's are associated with these tt's, and hence with one another. In exactly the same manner we can prove that with each p there is associated at least one it, and, if two or more /o's be associated with one another, at least as many tt's are associated with these p's, and hence with one another. I76 THE REALM k(i). Hence considering, as we always shall, two associated factors as the same, the two representations are identical ; that is, if in the one representation there occur e factors associated with a certain prime, there will be in the other representation exactly e factors associated with the same prime. We can now evidently write every integer, a, of k(i) in the form a = «-!**/■ • • ■ 7T n en , where v lf v 2 , »••,*« are the unassociated prime factors of a, and c a suitable unit. Moreover, this representation is unique. Cor. 1. If a and (3 be prime to each other and y be divisible by both a and /?, then y is divisible by their product. Cor. 2. // a and (3 be each prime to y, then af3 is prime to y. Cor. 3. // a be prime to y and a/? be divisible by y, 13 is divis- ible by y. We have seen that the divisibility of n[a] by n[fi] is a neces- sary condition for the divisibility of a by /?. We shall now show that it is only when either a or J3 is a rational integer that the condition is also sufficient. Let a m fc***^ - */*, = %??(>? - f? be representations of a and (3 as products of powers of their dif- ferent prime factors, rj a and rj^ being units. From n[a]=m • n[/3], where m is a positive rational integer, it follows that *i JL 2 ' L k '4 ''2 "k In "1 "2 ri r\ rz ri > from which we see that each prime, p i} of the set p 1} p 2 , '•',p l is associated with one of the 7r's or with one of the tt"s, say vj or «■/, and that ri 3> pj. In order that a may be divisible by /? we must have every p associated with an unaccented ?r, which will not be in general the case. When, however, a is a rational integer we have a = a', and this condition is satisfied, and hence /? divides a. If be a rational integer it is easy to see likewise that, when n[a] is divisible by n[(3], a is divisible by (3. THE REALM k(i). I 77 §9. Classification of the Prime Numbers of fe(i). Every prime, ir, of k(i) divides an infinite number of positive rational integers; for example, u[tt] and its multiples. Among these positive rational integers there will be a smallest one, p, and p will be a rational prime number, for if p be not a prime, that is, if p=p 1 p 2 , it would divide either p x or p 2 , and hence p would not be the smallest rational integer that -n divides. In order, therefore, to find all primes of k(i) we need only examine the divisors of all rational prime numbers considered as integers of k(i). Moreover it is evident that no prime of k(i) can divide two different rational primes, for then it would divide their rational greatest common divisor, I, and hence be a unit. Therefore every prime of k(i) occurs once and but once among the divisors of the rational primes considered as integers of k(i). We have seen already that there are rational primes, as 3, which are also primes of k(i), and other rational primes, as 5, which are factorable in k(i). Denoting then by p the smallest rational prime that it divides, we have p = ira, 1) and hence p 2 = n[ir]n[a]. We have then two cases . f*H=/>, .. f *[«]**#*, \n[a]=p. ' \n[a] = i. i. From n[ir] =inr'==p and 1) it follows that a = 7r'. If tt = a + bi, we have then p = a 2 + b 2 . Assume p =f= 2 ; then either a or b must be odd and the other even and therefore /> = i, mod 4. Hence when a positive rational prime other than 2 is the product of two conjugate primes of k(i), it has the form 4n + i- When p==2, we have 2=(I+*)(I— f), 12 I78 THE REALM k(J). and hence 2=*{i — i) 2 ; that is, 2 is associated with, and hence divisible by, the square of a prime of k (i) . ii. Since n[a] =»I, a is a unit and hence p is associated with the prime ir; that is, p is a prime in k(i). Hence a rational prime p is either a prime of k(i) or the product of two conjugate primes of k(i). When p is a prime of the form 4W — 1 it is always a prime in k(i), for we have seen that p is factorable into two conjugate primes of k(i) only when it is 2 or of the form 4n + 1. To prove now that every rational prime of the form 4W + 1 can be represented as the product of two conjugate primes of k(i) we observe that from £s==I, mod 4, it follows that the congruence .ar 2 = — i,mod£, has roots. Let a be a root. Then a 2 = — 1, modp, and hence (a -\- i) (a — ») aso, mod p. Since a-\-i and a — i are integers of k(i), the integer p, if a prime of k(i), must divide either a + t or a — i. This is how- ever impossible, for from a ± i=p(c + di), where c + cfo' is an integer of k(i), it would follow that pd=± 1, which can not hold since p and d are both rational integers and p > I. Hence /> is not a prime in &(*)> and since the only way in which a rational prime can be factored in k(i) is into two conju- gate prime factors, p is factorable in this manner. Collecting the above results, we see that the primes of k(i) may be classified in the following manner, according to the rational primes of which they are factors. 1 ) All positive rational primes of the form 4%-\- 1 are factor- able in k(i) into two conjugate primes, called primes of the first degree. THE REALM k(i). 1 79 2) All positive rational primes of the form 411 — 1 are primes in k(i), called primes of the second degree. 3) The number 2 is associated with the square of a prime of the first degree. It will be observed that the norm of every prime tt of k(i) is a power (first or second) of a rational prime and that the degree of tt is the exponent of this power. Moreover, we notice that 2 is the only rational prime that is divisible by the square of a prime of k(i) ; for, if this were true of any other rational prime of the form 4» + 1, we should have tt associated with tt', and hence a -j- bi = a — bi, — a + bi, b + ai or — b — ai, which give a = o, b=^o, or a = =%=b, all of which are seen to be incompatible with p = a 2 + b 2 . § 10. Factorization of a Rational Prime in k(i) determined by the value of (d/p). The rational primes may be classified with regard to their factorization in k(i) in the following manner: 1) Those of which the discriminant is a quadratic residue are factorable into two conjugate primes in k(i), called primes of the first degree. For (d/p) = i implies p = ^n-\- 1, since d = — 4, and we have seen that all rational primes of this form are thus factorable in k(i). 2) Those of which the discriminant is a quadratic non-residue remain primes in k(i), called primes of the second degree. For (d/p)= — 1 implies p = ^n + 3, and we have seen that all rational primes of this form remain primes in k(i). 3) Those which divide the discriminant {expressed symbol- ically by (d/p) =0) are associated with the squares of primes of the first degree in k(i). Evidently 2 is the only rational prime which divides the dis- criminant of k(i) and we have seen that 2 = i(i — i) 2 . The following table expresses the above results : ©=■•>= l80 THE REALM k(i). 3) (jh°' p= Ex. Show that, if a, =a-\-bi, be any integer of k(i), such that a and b have no common rational divisor, and c be any rational integer divisible by a, then c is divisible by n[a]. §11. Congruences in k(i). Exactly as in the case of rational integers, we say that two integers a, (3, of k(i) are congruent with respect to the modulus, /x, if their difference be divisible by fi, and write a = p 3 mod /a. The laws of combination that were proved for congruences in R hold here. We can now divide all integers of k(i) into classes with respect to a given modulus, li, putting two integers in the same class or different classes, according as they are or are not congruent to each other, mod /x. We shall show that for any given modulus n there will be n[fi] such classes. To do this we shall need the following theorem : Theorem 5. There exist among the multiples of any integer fi, of k(i) two, t 1 , = oo» 1 , i 2 , = bw 1 + C(o 2 , such that every multiple of p can be expressed in the form where a, b, c, l lt l 2 are rational integers and <d 1} w 2 is a basis of k(i). Suppose all multiples of li to be written in the form 1 = a 1 b) 1 -j- a 2 <n 2 , and consider those in which a 2 =|=o. Among them must be some in which a 2 is smaller in absolute value than in any of those remaining. Let t 2 , =bo> 1 -\~co) 2 , be one of these-; then c Will divide the coefficient a 2 in every multiple of fi ; for, if this be not the case, x This indicates that p is unfavorable in the realm under discussion. THE REALM k(i). l8l let /?, = b 1 <o 1 + c x w 2 , be a multiple of /x such that c x is not divisible by c, and let d be the greatest common divisor of c and c 4 . There exist two rational integers e, e lf such that ec + e x c x = rf, and hence y = £i 2 + e iP = ( *& + e J>i ) <"i + ^2 is a multiple of /x in which a 2 is less in absolute value than c, but not o. But this is contrary to our original hypothesis. Hence we have a 2 = l 2 c, where L is a rational integer, and hence t — l 2 i 2 s= (<*, — / 2 & ) Wj. Consider now those multiples of /* in which a 2 =:o, but a x =%=o. There will be some among them in which a x is less in absolute value than in any of those remaining. Let tp =a<o lf be one of these. It is seen as above that a is a divisor of the coefficient a x in every multiple of p. in which a 2 = o, ^=4=0. We have, therefore, since (a^ — l 2 b)<o x is a multiple of fi belonging to this class, t — / 2 t 2 = ( a x — l 2 b ) <o x = l x i x , where l x is a rational integer, and hence l = 1 1 l 1 j t1 2 i 2 . Any pair, fi x ,ix 2f of multiples of p, such that every multiple of fi can be written in the form m xf i x + m 2t i 2 , where m x ,m 2 are rational integers, we call a basis of the mul- tiples Of fl. The pair of multiples of p, a<o x , b<o x + c&> 2 , selected as above, and in which in addition a and c are positive, is called a canonical basis of the multiples of p. Theorem 6. // p x , fi 2 be a basis of the multiples of p, the necessary and sufficient condition that H* = a x fi x + a 2f i 2 , M 2 * = £i/*i + & 2 /*2> 1*52 THE REALM k (t) . zuhere a x , a 2 , b lt b 2 are rational integers, shall be also a basis of the multiples of /jl is a 1 a 2 b< b n = ± i The proof of the theorem is the same as that of Th. 2. Theorem 7. // fi x = a 1 (a 1 -J- a 2 (ti 2 , fi 2 = b 1 o> 1 + b 2 <o 2 , be any basis of the multiples of ft, then b t b 2 "*W; It is evident from the last theorem (see proof of Th. 2) that the absolute value of the determinant a x a is the same for every set of basis numbers of the multiples of p. Hence we need only determine its value for some particular basis. The integers fi = a x -f- a 2 h fxi = — a 2 -f- a ± i, constitute a basis of the multiples of p, and Hence the theorem is proved. Theorem 8. // fi be any integer of k(i), the number of num- bers in a complete residue system, mod fi, is n[fi]. Let cua lt b(d x + c<»2 be a canonical basis of the multiples of fi and consider the system of integers ( u = o, 1, "-,a — 1, U<a t -\- Vu) 9 < 11 2 { v=pO, I, ••-,£ — I, I) which are evidently ac, =n[fx], in number. We shall show that the integers 1 ) constitute a complete residue system, mod fi. THE REALM k(i). 1 83 First, each of them is incongruent to all the others, mod p., for if fijttj -\- v x w 2 , n 2 v> x -\- v 2 <o 2 be any two of them, and u 1 (a 1 -\- v-^2 sa u 2 (o 1 -f- v 2 (o 2 , mod fi, then (u 1 — u 2 ) <*>! + (y x — z/ 2 )<o 2 =o, mod /a, and hence, since c is the greatest common divisor of the coeffi- cients of w 2 in all multiples of fi, v x — z/ 2 ==o, mode. But v t and v 2 are both less than c; hence v x = v 2 . It follows that u 1 — w 2 = o, mod fi, and hence, since a is the greatest common divisor of the coefficient of (i) t in all multiples of fx in which the coefficient of w 2 is o, u i — u 2 — °> m °d a - But u x and w 2 are both less than a ; hence u x = u 2 . Thus w^i -(- v 1 o) 2 = u 2 o) 1 -\- v 2 <o 2 , and the numbers i) are seen to be incongruent each to each, mod fi. Moreover, every integer of the realm is congruent to one of the integers i), mod ll. For, let 0) = f 1<ttl + t 2 <o 2 be any integer of k(i), and let t 2 = mc + r 2 , where m and r 2 are rational integers and r 2 satisfies the condition o g r 2 < c. Also let f t — mb=na-\-r x , where n and r, are rational integers and r x satisfies the condition ogr 1 <ia. Then t x v x -f- t 2 o) 2 = (mb -f- na + r i) w i + ( mc + r 2 ) <a 2 = no*} + m{bu x + c<o 2 ) -f- r lWl + r 2 w 2 ; 184 THE REALM k(i). and hence t 1 o) 1 + 1 2 co 2 m r^! -j- f 2 <o 2 , m °d /a, where r^ + r 2 a> 2 is one of the integers 1 ) . Hence every integer of the realm is congruent, mod /a, to one and but one of the integers 1). The integers 1) constitute, therefore, a complete residue system, mod fi, and being n[p] in number the theorem is proved. We can construct a complete residue system for any modulus, fi, by means of the method employed in the above proof. Taking 1, i as a basis, we let n = m(p + qi), where m is the largest rational integer that divides p, p and q being consequently prime to each other. It is easily seen that m(p 2 -\-q 2 ) is the rational integer of smallest absolute value divisible by fi; that is, a = m(p 2 + q 2 ). Since ac=n[fi] — m 2 (p 2 + q 2 ), we have therefore c = m. q Hence the n[fi] integers tt = o,i, ••■,m(p 2 + q 2 ) — 1, v = o, 1, '-' } m — 1, is a complete residue system, mod /*. Ex. Let fi = 3 + 6*' = 3(1 + 2/). Then m = 3, a =15, c = 3. The following 45 integers constitute a complete residue system, mod 3 + 6V, 01 234 5 67 i 1 + i 2 + i 3 + » 4 + 1 5 + t 6 + * 7 -f « 2f 1+2* 2 + 2* 3 + 2* 4 + 2* 5 + 2* 6 + 2J 7 + 2* 8 9 io ii 12 13 14 8 + * 9 + * 10 + * 11 +* 12 + * 13 + * 14 + *. 8 + 2* 9 + 2.1 10 + 2* 11+ 2* 12 + 2/ 13 + 2* 14 + 2*. We can thus obtain a complete residue system with respect to any modulus by means of the method employed in the above theorem. There are two important special cases which deserve mention. i. If fi = p + qi, where p and q have no common divisor, the u + vi, i THE REALM k(i). 1 85 integers i, 2, •••, p 2 + q 2 , = n(n), form a complete residue sys- tem, mod fi. ii. // ix = m, a rational integer, the m 2 integers - I y = o, 1, •-., \m\ — 1, form a complete residue system, mod m. Ex. 1. Prove i and ii without making use of Th. 8. Ex. 2. Show that a as 13, mod 7, implies a' ^ /?, mod 7'. All integers belonging to the same residue class, mod /*, have with fx the same greatest common divisors ; for from a = (3, mod fi, it follows that a = fi-\-vfi, and hence every common divisor of (3 and fi is also a divisor of a and every common divisor of a and ^ is a divisor of (3. In particular, if one number of a residue class be prime to the modulus, fi, all other numbers of the class are prime to pu A system of integers incongruent each to each with respect to a given modulus, /x, and prime to ^ is called a reduced system of incongruent numbers, mod /a, or a reduced residue system, mod /*. Thus the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 constitute a complete system of incongruent numbers, mod 1 +3*, and 1, 3, 7, 9 con- stitute a reduced system to the same modulus. § 12. The ^-Function in k(i). Just as in R, we understand by <£(/*)> where ^ is an integer of k(i), the number of integers in a reduced residue system, mod p. We have 0(c) = 1, where c is any unit of k(i), and, as may be easily seen, <f>(Tr)=n[Tr] — 1, where tt is a prime of k(i) ; for example, <f>(2 -f-f) =4, since 1, 2, 3, 4, 5 constitute a complete residue system, mod 2 -f- i, and all these integers except 5 are prime to 2 -f- i. Likewise <Ki+3*')=4> I 86 THE REALM k(i). since I, 2, 3, 4, 5, 6, 7, 8, 9, 10 constitute a complete residue sys- tem, mod 1 -\- $i, and of these integers only 1, 3, 7 and 9 are prime to 1 + 3*. To get a general expression for <j>(fi) in terms of n, we may employ any one of the three methods used to obtain the corre- sponding expression in R. We shall sketch the proof briefly, following the third method used in R (see Chap. Ill, §4). The completion of this and the two remaining proofs will serve as exercises. Theorem 9. // a = /3y, where /? and y are any integers of k(i), there are in a complete residue system, mod a, exactly w(y) numbers that are divisible by /?. Let Yi>y 2 > •••>yn( V ) 1) be a complete system of incongruent numbers, mod y. The num- bers Pyi,Py**~->Py*to) 2) are incongruent, mod a, for if /tyfc = /tyi> mod a, then y fc ==y 4 , mody, which is impossible. Moreover, every integer (38, divisible by /? is congruent to some one of the numbers 2), mod a; for 8 is congruent to some one, say yi, of the numbers i), mod y, and from h^yi, mody, it follows that £8 = /^, mod a. Since, also, every integer congruent, mod a, to one of the num- bers 2) is divisible by /? (see § n and Chap. Ill, § 1, ix), and the numbers 2) are n(y) in number, there are in every complete residue system, mod a, exactly «(y) numbers that are divisible by/?. Theorem 10. // tt be any prime of k(i), ^ (Tm)=wM ( I __L_) THE REALM k(i). I 87 From the last theorem we see that among the n[ir m ] numbers of a complete residue system, mod ir m , there are exactly ^[tt" 1-1 ] that are divisible by tt, and hence n[ir m ] — n[7r wl ~ 1 ] that are prime to tt w ; that is <£O m ) =w[7r m ]( I — - i r— V \ »M / To derive the general expression for </>(/*) we have now to prove the theorem for k(i) corresponding to Th. 4, Chap. III. Theorem ii. // /i lt /*,, • • -,fx 8 be integers of k(i) prime each to each <£Oi/*2 •••/ a «) =^(^1)^(^2) •••<£0*«)- Ex. We have — 3 + n«= (1 + 30 (3 +2t)f where 1 + 3* and 3 -{- 2/ are prime to each other. Hence <t>{— 3+ lit) =0(1 + 300(3 + 20 =4- 12 = 48. The proof of this theorem depends directly upon the following theorem which can be proved exactly as in R (Chap. Ill, § 14) : Theorem 12. // /x = ^ 1 /x 2 ---^« where p x ,ix 2 , '-,n 8 are integers of k(i) prime each to each, and if a 1} a 2 , ••■,a 8 be any integers of k(i), there exist integers, <o, such that co ae cl x , mod fi x , to ^ ol 2 , mod fx 2 , • • • , w ^ a s , mod /a s , and all these integers are congruent each to each, mod p. More- over o> = a 1 ^ 1 + a 2 /3 2 + • • • + a 8 p8, mod ft, where Pi = 1, mod /xj, and /?* ==0, mod ^ • • • i*>i- 1 fj>i +1 • - • /**, *= 1,2, • • •, ^. We can now obtain easily the general expression for <f>([i), n being any integer of k(i). Theorem 13. If p be any integer of k(i) a)td 7^, 7r 2 , — ',tt 8 the different prime factors of jx, then fG0*=»M ( i -n^)( l -T L TV--( i --tM- Let 1 88 THE REALM k(i). By Th. ii we have *00 = <*>Oi ei )4>U 2 e2 ) •••♦(«/»), from which by Th. 10 it follows that +<*> = n[ ^ ,] ( ' ~spb ) wM ( ' " sib) -^']( I - ) T[i7]) and hence that Ex. We have — 201 — 43*' = (i + (2 + i)*(3 + 2O 2 , and hence <t> (— 201 — 43*) = » ( _ M _^( I __^)(,__i R )...(,__ f _J : ^) > = 42250 • I • f • H , = 15600. Theorem 14. // B 19 8 2 , • • ♦, 8 r fo ffo different divisors of n, then 1, r For proof see corresponding theorem in R (Chap. Ill, Th. 6). Ex. We have — 3 + ni = (1 +0 (2 + •) (3 -f 2.1). The different divisors of — 3 + 11* are 1, 1 + 1, 2 + t, 3 + 2 ^ * + 3*> i + 5*> 4 + 7** and — 3 + 11*, and for these the corresponding values of are 1, 1, 4, 12, 4, 12, 48, 48, whose sum is seen to be 130, = n[ — 3 + 111]. § 13. Residue Systems Formed by Multiplying the Numbers of a Given System by an Integer Prime to the Modulus. Theorem 15. // j^,/^, •••,**»[*] be a complete residue system, mod fi, and a any integer prime to /*, then a^, a/x 2 , • •*,a/*n[ M ] is also a complete residue system, mod p. The integers a^, a/x 2 , • • • , a/*„[>] are incongruent each to each, modju, for from OLfii^dixj, mod//., THE REALM k(i). 1 89 it would follow that, since a is prime to p, fii^fjij, mod jx, which is contrary to the hypothesis that fa, p 2 , • • • , p n [^ form a complete residue system, mod p. The integers a/x, 15 ap 2 , • • •, ap n [^ are, moreover, n[fi] in number. They form, therefore, a complete residue system, mod p. Cor. // p lt p 2 , — ' , p4>(m) be a reduced residue system, mod p., and a be prime to p, then ap 1} ap 2 , • - • , ap^^) is also a reduced residue system, mod p.; for ap ly ap 2 , • '*,ajp+(») are incongruent each to each, mod ^, prime to p and <£(/x) m number. §14. The Analogue for k(i) of Fer mat's Theorem. A theorem analogous to the generalized Fermat's theorem for rational integers can be proved for the integers of k(i) ; that is, Theorem 16. // p be any integer of k(i) and a any integer prime to p, then a*** 3 = i, mod^. Let a ly a 2 , •••,a < / >(tt ) be a reduced residue system, mod p.; then aa lf aa 2 , •••,aa^ (M) is also such a system (Th. 15, Cor.). Since aa x ,aa 2 , ••-,aa^ (|1 ) and a lt a 2i •••,a^ (M) are both systems of this kind, each integer in the one system must be congruent, mod p., to one and only one integer in the other sys- tem, though perhaps in a different order; that is, aa 2 = a k aa <tt ^)=a k<i)(ilx) . mod p. Hence a*Ma x a 2 -- a* (M) z=a kl a ki ••• a fc ^ (|0 , mod/*, and since a x a 2 • • • a$ (M) = a kl a ki - - • a fc</>(M) , 190 THE REALM k(i). and is prime to /a, we have a *(n)=== Ij mod/*. Cor. 1. If -rr be a prime and a any integer not divisible by it, then a »[T]-i == lf m0( j w This is the analogue of Fermat's Theorem. Cor. 2. If tt be a prime and a any integer of k(i), then a ra[flrl ==a, modTr. Ex. 1. Let 7r=i + 2«, and a=i + i; then (1 + Q+CM-iOpBi 1, mod 1 + 2f, or (i-\-i)* = — 4=1, mod 1 +2*. Ex. 2. Let fi= 1 + 31 and a = 3 ; then 3<*>(i+3i)^ Xj mo( j j _j_ £ or 81 ^ 1, mod 1 + 31. Ex. 3. If a and /* be any two integers of k{%) and a = cci8, n = f^S, where 8 is the greatest common divisor of a and /*, show that the necessary and sufficient condition for a+GO+i^o, mod /*, is that n x be prime to 5. § 15. Congruences of Condition. The remarks at the beginning of §9, Chap. Ill, apply equally to congruences in k{%) 9 and the theory of congruences of con- dition in k(i) can be developed in exactly the same manner as in R. In k(i) the coefficients of the polynomials are any integers of k(i). With this change we can show that a polynomial in a single variable x can be resolved in one and but one way into prime factors with respect to a modulus which is a prime of k(i), and upon this theorem build a theory for congruences in one unknown just as in R. The theories of power residues, binomial congruences and in- dices may be developed similarly for the integers of k{t). THE REALM k(i). I9I § 16. Two Problems. We shall now discuss briefly two problems which are of interest in the theory of numbers, the first being especially famous. They can be solved without making use of numbers other than those of R, but their solution is greatly assisted by the introduction of the realm k(i). Problem i. To represent a rational prime as the sum of two squares. 1 — Let p be a rational prime and suppose the desired rep- resentation possible. Then p = a 2 + b 2 , and hence p = (a + bi) (a — bi) ; that is, the representation is possible when and only when p is the product of two conjugate primes of k(i). Hence i. No prime of the form 411 -f- 3 can be represented as the sum of two squares, since a prime of this form is a prime in k(i). ii. The number 2 and every prime of the form 411 -{- 1 can be represented as the sum of two squares. Moreover, this representation is unique, for if we have two dif- ferent representations p = a 2 + b 2 and p = a 1 * + b 1 2 , then p=(a-\-bi)(a — bi) and p= (a x + bj) (a x — bj) ; that is, p would be factorable in two different ways into prime factors in k(i), which is impossible. Hence 2 and every prime of the form qn -f- 1 can be represented in one and only one way as the sum of two squares, but no prime of the form 411 -f- 3 can be so represented. Problem 2. To represent any positive rational integer, m, as the sum of two squares. Let m = p x p 2 • • • pr-qi*^** - • • qJ; where p lt p 2 , '--,pr are rational primes of the form 411 -j- 1 or 2, 1 Fermat: Works, Vol. I, p. 294. For solution of this problem without the aid of k(i) see Dirichlet- Dedekind: §68; also Mathews: §91. 192 THE REALM k(i). two or more of which may be alike, and q lt q 2 , m ",q8 rational primes of the form 4W + 3, that are all different from one another. If the representation be possible, m = a 2 -f- b 2 ; and hence m = (a -f- bi) (a — bi) . The representation is therefore possible when and only when we can factor m into two conjugate factors in k(i). The necessary and sufficient condition for this is that all the fs be even, in which case we have, if Pi =:= 1T\K\ , P2 = 7T 2 7r 2 , ' ",pr = TTrTTr , m = (»!», • • • atfe*^,"* ■ ■ ■ q s u '*) X Hence if a positive rational integer, m, contain a prime factor of the form 4n-\- 3 an odd number of times, m cannot be repre- sented as the sum of two squares. In all other cases the repre- sentation is possible. Moreover, supposing the factorization 1) to be possible, it can be effected in general in several different ways, as for example, m = (»>, . . . wrtflfiqf* • • • q. u ' 2 ) X and since each of these factorizations yields a different represen- tation of m as the sum of two squares, the problem can be solved in exactly as many different ways. If m~2 n p x e ip 2 e *-'pr er q 1 H q 2 t2 '--q Si u , where the p's are primes of the form 4n -\- 1, all different, the q's primes of the form 4 n + 3> an d th e ? s oil even, then, if N be the number of different ways in which m can be represented as the sum of two squares, we have N = i(e 1 + 1) (e 2 + 1) - • • (e r + i) or\(e x + 1) (e 2 + 1) • * • ( e r + 1) + i according as some or none of the e's are uneven. (See Gauss: Disq. Arith., V, 182.) Ex. 65 = 13 • 5 = (1 + 2O (1 — ») (2 + 3O (2 — 31), = [(i+ 2 (2 + 301 [0 -20(2-30], = (-4 + 70(-4-70=4 2 + 7 2 , or as [(i + 20(2 — 30H(i— 20(2 + 30], = (8 + 0(8-0 =8 2 + i 2 . Thus 65 can be expressed in two ways as the sum of two squares. THE REALM k(l). 1 93 § 17. Primary Integers of k(i). When an integer, a, plays the role of divisor it is unnecessary to distinguish between its associates. This is, however, not the case when a is combined with other numbers by the operations of addition or subtraction. For example, when a is the modulus of a congruence we may consider a to be any one of its asso- ciates, but when a is a coefficient some particular one of its asso- ciates must be designated. This distinction between the associates of a is the same as that made in the rational realm between a and — a. There, for example, the quadratic reciprocity law is given for positive primes, since although we have always (0-&> we do not have in general is)-m An integer so singled out from its associates according to some prescribed rule is called a primary integer. This rule of selection should evidently be such that the product of any two primary integers is primary; that is, if a and p be the integers selected as primary from a, — a,ia, — ia- and p, — P, ip, — ip, respectively, then ap should be the integer that ac- cording to the same rule should be selected as primary from ap, — ap,iap, — iap. Gauss gives two rules of selection, both of which obey the principle just enunciated. The first rule is based entirely upon this principle, the second partially. Gauss makes use of the sec- ond rule and this one will now be described. The rule will be given here without employing the above men- tioned principle, and will then be shown to obey it. We first divide the integers of k(i) into two classes according as their norms are odd or even, those of the first class being called odd integers, those of the second class even integers. 1 1 Bachmann : Die Lehre von der Kreisteilung, p. 152. 13 194 THE REALM k(i). If n[a-\- bi], = a 2 + b 2 , be odd, it is evident that either a or b is odd, the other even. If n[a + bi] be even, a and & are both odd or both even. Every prime of k(i) except I + i is evidently an odd integer. Since I -\-i and I — i are associates, it is evident that n[a] divis- ible by 2 is not only a necessary but a sufficient condition that a shall be divisible by I -f- i. We see, therefore, that a necessary and sufficient condition for an integer of k(i) to be even is that it shall be divisible by i -f- i- The selection of one of the four associates of an integer is now made as follows. Considering first only the odd integers of k(i), we have the following rule: That number x -\- yi of the four associated odd integers a -\- bi, — a — bi, — b + ai, b — ai i ) is singled out as primary in which we have simultaneously either x*s~ i ; y = o ^ V , mod 4, 2) or x = — i ; y = 2 ) where x denotes the real part and y the coefficient of i. That one and only one such integer exists in the group i) is shown as follows. Since a -\- bi is an odd integer, a and b can neither be both odd nor both even. Suppose a even, b odd. Then one of the integers, b or — b, is of the form 4» + i, the other of the form ^n — i. If now a = o, mod 4, that one of two integers, b — ai, — b -j- ai, will be primary in which the real part has the form 411 -f- 1 • If a = 2, mod 4, that one of the integers, b — ai, — b -f- ai, will be primary in which the real part has the form 4^ — 1. It is evident in both these cases that none of the remaining associates satisfy the conditions. Similarly we see that when a is odd and b even, one and only one of the four associates 1) satisfies 2). THE REALM k(i). 1 95 If a be a rational integer, that one of the integers, a, — a, is primary which has the form 411 + 1 . The negative rational primes prime in k(i) are thus seen to be primary. Two conju- gate odd integers are evidently either both primary or both non- primary. It can be easily shown that the above rule of selection is equivalent to the following : That one of four associated odd integers is primary which is congruent to 1, mod 2 + 21. 3) Ex. Of the four associated odd integers 9 -f- 121, — 9 — 12*, 12 — gi, — 12 -f- gi, 9 +121 satisfies the conditions 2); for we have 9^1 and 12^0, mod 4. Hence 9 + 12* is primary. We also see that 9 + 12* = 1, mod 2 -\- 22. It is easily seen that 9 -f- 12* is the only one of its associates which satisfies the conditions 2) or their equivalent 3). Since every prime of k(i) except I + * is an odd integer, we can now distinguish between the associates of every prime except 1 -}- *. In the case of 1 + i we may take any one of its associates, say 1 + i as the primary one. The primary primes of k (t ) whose norms are less than 50 are I +t, — 1+21, —I—2i, —3, 3 + 2/, 3 — 21, I +4*, I—4/, — 5 + 2«, — 5 — 2/, — 1 -f 6i, — 1 — 6i, 5 + 4h 5—4*', — 7- Remembering that a necessary as well as sufficient condition for an integer, fx, to be even is that it shall be divisible by 1 + i, we can distinguish between the associates of /a by taking that one as primary which when written in the form ( 1 -\-i) n v has the factor v, which is an odd integer, primary. We shall now show that the product of two odd primary integers is a primary integer. Let a, = a -j- hi, and j3, = c + di, be any two odd primary in- tegers. Then one of the following cases must occur. mod 4, 1. a= 1 11. 1 1 a==- 11. — 1 IV. ass— 1 * b = o b = b = 2 b== 2 c== 1 c = - - 1 c^ 1 C = I d = o d = 2 d^ dz= 2 J I96 THE REALM k(i). and afi=(ac — bd) -f- (ad -\- bc)i = e + fi, gives one of the following corresponding cases : i. ii. iii. iv. V , mod 4. Mi' / = f&B 2 /eee 2 Hence a($ is always an odd primary integer, if a and /? be odd primary integers. This may be shown more simply by means of the condition 3). From this it follows at once that the product of any two pri- mary integers is primary. We may now express the unique fac- torization law for the integers of k(i) as follows: A primary integer can be resolved in one and only one way into a product of primary prime factors. The term primary integer is generally taken to mean what is here called an odd primary integer. § 18. Quadratic Residues and the Quadratic Reciprocity Law in k(i). 1 If a and fx be any integers of k(i) prime to each other, we say, as in R, that a is a quadratic residue or non-residue of /u. accord- ing as the congruence x 2 = a, mod fi, has or has not roots. Ex. 1. The congruence x* as 1 -f- i, mod 1 — 21, has the roots ± 2 ; for (± 2) 2 = 1 + i, mod 1 — 2.1, since 4 — ( 1 + *) =3 — i= ( 1 -j- *) ( 1 — 21) . Hence 1 -f- i is a quadratic residue of 1 — 2fc Ex. 2. On the other hand the congruence x 2 s 3, mod 1 — 2%, has no roots, for substituting the integers ± I, ± 2, of a reduced residue system, mod 1 — 2%, we have \ , mod 1 — 2i. 4^3 j x See Gauss: Theoria Residuorum Biquadraticorum, §§ 56-60; Works. Vol. 2, pp. 126-130. THE REALM k(i). \ gf Hence 3 is a quadratic non-residue of I — 2f. The theory of quadratic residues can be developed for k(i) along lines so nearly identical with those for the same subject in the rational realm that only the briefest outline will be given here. We have, as before, two questions to answer: first, what in- tegers are, and what are not, quadratic residues of a given modu- lus; second, of what moduli is a given integer a quadratic residue and of zvhat moduli is it a non-residue ? The first question can be easily answered. The second is much more difficult. We shall confine ourselves in what follows to the case where the modulus is a prime r. We observe first that every odd integer of k(i), that is, every integer prime to I + h is congruent to I, mod I + h an d hence is a quadratic residue of I -f- i. For 7r, an odd prime, we have the following theorem, the proof of which is like that of the corresponding theorem for rational integers (Chap. IV, Th. i). Theorem 17. The necessary and sufficient condition that a shall be a quadratic residue of ir is that Ex. 3. Let tt= 1 — 2.x, a= 1 + i. We have n|>]-l (1 + 2 = (1 + 2 = 2t as 1, mod 1 — 2i. Hence 1 -f- 1 is a quadratic residue of 1 — 21. Ex. 4. Let 7T ss 1 — 21, a = 3. We have »[ir]-l 3 2 =3 2 = 9^i, mod 1— 2»* Hence 3 is a quadratic non-residue of 1 — 21. These results are con- firmed by Ex.'s 1 and 2 above. Cor. The integer a is a quadratic residue or non-residue of ir according as we have a 2 =1 or — 1, modir. 1 Let now, as in the rational realm, the symbol (a/n) have the 1 See Chap. IV. Th. 1. Cor. 1. I98 THE REALM k(i). value 1 or — 1 according as a is a quadratic residue or non-residue of sr, we have I - ) m a 2 , mod it. The symbol (a/V) obeys the following laws i. If « = /?, modTr, te)-(f-)' a Sine. (=-!)_ ('-) _,, -- (v)-(t) (:-)=(,-)■ iv. Since (l)-(^)--(;)-(i) it follows that ©-(-«)-(i)-(-:). v. Since y 2 = a, modir, implies y' ^=ol', mod 7/, we have i)-6) Every integer a can be written in the form a = i r (i+i) 8 p lP2 -- P n, where r = o, 1 , 2 or 3, s = o or a positive integer, and p x , p 2 > * * • > P* are odd primary primes. We have then (=)-a)-(^m)(?)-fe), and the determination of the value of (a/v) is seen to be resolved into- the determination of the values of THE REALM k(i). 1 99 aM^'MD where p is an odd primary prime. The close similarity between this resolution of our original problem into simpler ones and the corresponding case in the rational realm should be noticed. Theorem 18. The unit i is a quadratic residue or non-residue of a prime it according as n[?r] is of the form 8m -\- i or 8 m + 5- If ir be a prime of the first degree, w[tt] is a positive rational prime of the form 4k + 1, and hence either of the form 8w + 1 or 8m + 5. If 7T be a prime of the second degree, n[ir] is the square of a rational prime of the form 4k + 3, and hence of the form 8m -|- 1. We have from Th. 17 2 , mod 7T, u)- (i\ »r»]-i and hence (-)==(— 1) 4 , mod ir, »[ir]-l or since ( _ ,j 4 = T or _ z (';)-<-> »[*]-! 4 But («[ir] — 1)/4 is even or odd according as n[ir] is of the form 8m + 1 or 8m + 5. Hence (*/*•) = 1 or — 1 according as n[ir] is of the form 8m + 1 or 8m + 5- We observe that i is a quadratic residue of all primes of the second degree. The solution of the same ques- tion for 1 + i is obtained by Gauss inductively as follows i 1 We find by means of Th. 17 that 1 + i ^ is a quadratic residue of the following primary primes — I + 21, 3 — 2f, — 5 — 21, — 1— 6/, 5+4*', 5 — 4*', —7, 7 + 2 ', — 5 + &, etc., and a quadratic non-residue of — 1 — 21, — 3, $-{-2i, I +4*, 1 — 4*, — 5 + 21,-1+61, 7 — 2/, —5 — 6/, —3 + 81', —3 — 8/, 5 + 8/, 5 — 8/, 9 + 4*, 9 — 4/, etc. *Th. Res. Biquad., Com. Sec, §58; Works, Vol. II, p. 128. 200 THE REALM &(*). Examining these series of primes we see that those in the first class are all such that a-\- &= i, mod 8, and those in the second class such that a + b bb — 3, mod 8. Hence it seems probable that I + ' 'is a quadratic residue or non- residue of an odd primary prime, a -\- hi, according as we have a + b = i or — 3, mod 8, one of which cases must always occur (see definition of primary integer). Since the quadratic character of an integer is the same with respect to all associates of tt, and in particular \a + bij \ — a — biJ we see that, if the above induction be correct, I + * is a quadratic residue or non-residue of — a — bi according as — a — & = — I or 3, mod 8, a + bi being an odd primary prime. Assuming the correctness of the above inductive reasoning, we have the following theorem : Theorem 19. If a-\- bi be a prime such that a is odd and b even, 1 -\-i is a quadratic residue or non-residue of a-\- bi, ac- cording as a-\-b=± 1 or ± 3, mod 8. This theorem may be proved by treating it as a special case of a more general theorem (Th. 22), which we shall consider in the next section. 1 To determine the value of ( . . ] we have only to remember that (^•) = (S) (seevabove )- 1 For an independent proof see Dirichlet, Crelle, Vol. XXX, p. 312. THE REALM k(i). 201 and hence since ( j. ) = i,whena+( — b) as ± i, modSt and = — i, when a + ( — &) ■■ dz 3, mod 8, we have V* + &7 = 1, when a — & = ± 1, mod 8, : — 1, when a — &=±3, mod 8. Ex. 1. Deduce the above criterion for the value of ( — ^— ) from Va-f-fr*/ the fact that (l=*\ = ( *A f-l±J^ . Ex. 2. Under what condition is \a + bi) \a + bij Gauss proceeds next to the consideration of the question: Of what odd primary prime moduli is a given odd primary prime a quadratic residue and of what a non-residue? The analysis em- ployed in the discussion of this question so beautifully exemplifies what can be accomplished in the theory of numbers by induction, this constituting, as Gauss says, 1 " the peculiar charm " of this branch of mathematics, that we shall give it in full. , The following is a free translation of §§ 59, 60, Commentatio Secunda, Theoria Residuorum Biquadraticorum. " Passing to the odd prime numbers, we find the number — 1 + 21 to be a quadratic residue of the moduli 3 -f- 2i, 1 — 4*, — 5 -\- 21, — i—6i, 7 — 2i, _3 + 8i, 5 + 8*, 5 — 81, 9 + 4, etc., but a non-residue of the moduli — 1 — 2i, — 3, 3 — 2», 1+4*, — 1 + 6t, 5 + 4h 5 — 4*', — 7> 7 + 2i, —5 + 6/, —5 — 61, — 3 — 81, 9 — 4*, etc. Reducing the moduli of the first class to their residues of least absolute value with respect to the modulus — 1 -f- 2i, we find these to be — 1 and 1 only ; for instance, 3 + 2* = — i, 1 — 4**2= — i, — 5 -|-2«= 1, — 5 — 212s — 1, etc. 1 Gauss: Works, Vol. II, pp. 152 and 157. 202 THE REALM k(i). On the other hand, all moduli of the second class are found to be congruent to either i or — i with respect to the modulus — I -f- 21. 1 But the numbers i and — i are themselves quadratic residues of the modulus — I -f- 2*, while i and — i are non-residues of the same modulus ; wherefore, so far as induction may be trusted, we obtain the theorem : The number — I -\- 21 is a quadratic residue or non-residue of the prime number a-\-bi according as a -j- bi is a quadratic residue or non-residue of — 1 + 2% itself, if a + bi be the primary one of its four associates, or more exactly if merely a be odd and b even. Moreover, from this theorem follow immediately similar theo- rems for the numbers 1 — 2i, — 1 — zi, 1 -f- 2.i. Since ( L ~ 2i \ = ( — l \ ( ~ 1 + 2i \ = (~ I + 2i \ \a + bi) \a + bi)\ a + bi ) \ a + bi /' we have (l=g) = (2 + «Y \a-{-biJ \i — 2%) Also f- 1 - 2 ^ = (-1±2\ = (JlZZ*L\ = ( ° + bi\ \ a + bi J \ a — bi ) \— 1+21 J \—i—2iJ and then as above (*+f\ = ( a + hi \ \a + bi) \i+2ij' Instituting a like inductive enquiry concerning the numbers — 3 or 3, we find that both are quadratic residues of the moduli 3 + 2*, 3 — 2f, — 1 + 6», — 1 — 6i, — 5 + 6i, — 5 — &, — 3 + Si, — 3 — Si, 9 -f- 4i, 9 — zji, etc., but non-residues of — 1 -f- 2f, — 1—2?; 1+4*, i— 4*, —5 + 2i, —s — 2i, 5 + 4*', 5—4^ 7 + 2*, 7 — 2*, 5 + 8*; 5 — 8*, etc. The former are congruent with respect to the modulus 3 to some one of the four numbers 1, — 1, i, — i; the latter, however, to some one of the four numbers 1 -\-i, 1 — i, — I +.*, — 1 — i. 2 1 It will be observed that 1, — 1, i, — -i constitute a reduced residue system, mod — 1 + 2*- 2 The numbers 1, — 1, i, — i, i-\-i, J — i> — *+** — 1 — * constitute a reduced residue system, mod 3. THE REALM k{l). 2C>3 The numbers i, — i, i, — i are themselves quadratic residues of 3, while I + i x — h — I + ** — * — * are non-residues. Induction teaches, therefore, that the prime number a + bi, supposing a odd, b even, has the same relation to the number — 3 as — 3 has to a -f- hi, in so far as the one is a quadratic residue or quadratic non-residue of the other, and like relations hold between 3 and a -\- bi. Applying a like inductive process to other prime numbers, we find in every case this most elegant law of reciprocity confirmed, and in the arithmetic of the complex numbers we are led to this fundamental theorem concerning quadratic residues : Theorem 20. 1 // a 1 -f- ° x i and a 2 -f- b 2 i be two prime numbers such that a x and a^ are both odd, b x and b 2 both even, then each will be a quadratic residue or each will be a quadratic non-residue of the other. But notwithstanding the extreme simplicity of the theorem its demonstration presents great difficulties, which, however, shall not delay us here, since the theorem itself is merely a special case of a more general theorem, which exhausts, as it were, the whole theory of biquadratic residues." We shall conclude this brief resume of the theory of quadratic residues in k(i) with the solu- tion of three examples. Ex. 1. To determine the quadratic character of 5 — 4*" with respect to the modulus 11 -\-6i. We have by the above theorem \ii+6i) \5 — 4*/ V5— 4//' 1 Since ( — )= ( )= ( )= ( ) it is not necessary to limit a and * to odd primary integers, but only to odd primary integers or those with their signs changed ; that is, integers of the form a -f- bi, where a is odd and b even. Expressed symbolically the theorem is t ax -f bti \ _ / ch -f b 2 i \ \ch -f- b 2 i) ~ \tf! -{- bii) Dirichlet gives a simple proof independent of the theory of biquadratic residues ; Crelle : Vol. IX, p. 379 ; also H. J. S. Smith : Works, Vol. I, p. 76. 204 THE REALM k(i). But 6+ioi= (i + 3 (i— 4*)- Hence &=£L)-(l±i)\i=gi • \5 — 4V \5 — 4*/ But f ^±A^ = 1, since 5+ (—4) =1, mod 8, (Th. 19), and by Th. 20 \5 — 4*/ \ 5 — 4* / \ 1 — 41 / \ 1 — 4* / V 1 — 4 l / Hence ( S ~ 4 \) = 1, Vii + 6t/ and the congruence x 2 ma 5 — 4/, mod 11+ 6i, has roots. Ex. 2. To determine the prime moduli of which 1 + 2.1 is a quad- ratic residue, and those of which it is a non-residue. Let a -f- bi be a primary prime and hence a odd, and b even. Then \a + bi) = \T+m) = VF+2?/' \T+2i)' Vj+2? ) or VT+JwJ' according as a -\- trims I, i, — 1, or — i, modi-f-2&. But (* ) = T) ( f\ = _ , (^1.) = I; and (j=L) =_ I( \l+2t/ \l+2*/ \I+2Z/ Vl+2*/ Hence 1 + 21 is a quadratic residue of a + fo when a + fee == 1 or — 1, mod 1 + 2*" and a quadratic non-residue when a -j- fo ^ * or — i y mod 1 -f- 2*. Therefore 1 -|- 21 is a quadratic residue of all primary primes included in the forms p(i-\-2i) ±1, 1) and a quadratic non-residue of all primary primes included in the forms /a(i +2/) ± i. 2) Ex. 3. To determine the prime moduli of which 3 -j- 6* is a quadratic residue, and those of which it is a non-residue. Let a -j- bi be a primary prime. We have ft±|ft = (— L-} fl±£\ We find as in the last example that ( fu ) =z h w ^en a + bi is a THE REALM k(i). 205 primary prime contained in one of the forms 3fi ± i, 3fi ± i, 3) and f , . ] = — i, when a + bi is a primary^ prime contained in one of the forms SH± (i-f-O, 3/*± (i— «). 4) Combining these with the results obtained in the last example, we see that 3 -f- 6i is a quadratic residue of all primary primes contained simul- taneously in the forms i) and 3), or simultaneously in the forms 2) and 4), and their associates. On the other hand 3 + 6* is a quadratic non-residue of all primary primes contained simultaneously in the forms 1) and 4), or simultaneously in the forms 2) and 3). These conditions may in each case be combined into a single one by Th. 12. § 19. Biquadratic Residues. It is impossible to leave the realm k(i) without a few words as to the history of the first treatment of these numbers from the point of view of the theory of numbers, marking as it did a distinct epoch in the development of this branch of mathematics. On the fifth of April, 1825, Gauss laid before the Royal Society of Gottingen a paper 1 upon the subject of biquadratic residues, a brief report 2 of which is given in the " Gottingische Gelehrte Anzeigen" for April 11, 1825. He remarks in this report that : " The development of the gen- eral theory which requires a most peculiar extension of the field of the higher arithmetic 3 is reserved for future continuation, only those investigations being taken up in this first paper which can be completely carried through without this extension," giving thereby a foretaste of a step which was to revolutionize the theory of numbers ; a step, however, the results of which he did not pub- lish until six years later. In this first paper Gauss defines a biquadratic residue as fol- ^heoria Residuorum Biquadraticorum : Commentatio Prima. Works, Vol. 2, p. 65. 2 Ibid., p. 165. 'Italics are the author's. See also H. J. S. Smith: Report on the Theory of Numbers, Arts. 24-36; Works, Vol. I, pp. 70-86, and Bach- mann: Die Lehre von der Kreisteilung, Vorlesung 12th. The reader is especially advised to consult Gauss' reports on his two papers and H. J. S. Smith's resume. 206 THE REALM k(i). lows : " An integer a is called a biquadratic residue of the integer p when there exist numbers of the form x* — a which are divisible by p, and a biquadratic non-residue of p when no number of this form is divisible by p" or we may say, as in Chap. Ill, § 34, that an integer, a, is a biquadratic residue or non-residue of an integer, p } according as the congruence x* — a = o, modp, has or has not roots. Limiting the investigation now to the case in which p is a posi- tive prime of the form 411 + 1 and a not divisible by p, all other cases being as he says reducible to this one, he separates all integers, a, not divisible by p, into four classes, according as fl i(p-D 53 1, f, — 1, or — j, mod p, where / is a root of the congruence f 2 -f- 1 s= o, mod p. Every integer of a reduced residue system, mod p, satisfies the congruence x?~ x — 1=0, mod p, 1 ) which may be written (**<p-i> — 1) O^- 1 * — /) (x^p-v + iX**-^ + /) =0, modp, 2) where /, — / are the roots of the congruence x 2 + 1 == o, mod p. Since the congruence 1 ) has exactly p — 1 roots, each of the four congruences into which 2) can be resolved has exactly \{p — 1) roots and the integers of a reduced residue system, mod p, are seen to fall into four classes, each containing J(/> — 1) integers, according as they satisfy the first, second, third or fourth of these congruences. The first class comprises those integers for which the congru- ence 1) is solvable; that is, the biquadratic residues of p (Chap. Ill, Th. 31) ; the third comprises those integers which are quad- ratic but not biquadratic residues of p ; the second and fourth classes are made up of the quadratic non-residues of p. THE REALM k(i). 20? We see, therefore, that, as Gauss remarks, all biquadratic resi- dues of p are also quadratic residues of p and all quadratic non- residues of p are also biquadratic non-residues of p ; but that not all quadratic residues of p are biquadratic residues of p. Gauss now divides the investigation, as in the case of quadratic residues, into two parts according as p or a is supposed given ; that is, ac- cording as we are to find what integers are biquadratic residues of a given prime modulus and what non-residues, or of what prime moduli a given integer is a biquadratic residue, and of what a non-residue. The first of these is elementary in comparison with the second and easily solved. Of the second part he treats three special cases, a = — I, a=±2, but does nothing with the general case. These three special cases, however, he fully discusses, remarking upon the exceeding difficulty of the cases a=±2. In this connection H. J. S. Smith says :* " The result arrived at in the case of 2 is that, if p be resolved into the sum of an even and an uneven square (a resolution which is always possible in one and only one way), so that p = a 2 -f- b 2 (where we may sup- pose a and b taken with such signs that a= 1, mod 4, b = af, mod />), 2 belongs to the first, second, third or fourth class according as \b is of the form 411, 4» + 1, 411 +2 or ^n -f- 3. "The equation p = a 2J r b 2 shows that p=(a-\-bi)(a — bi), or that p, being the product of two conjugate imaginary factors, is in a certain sense not a prime number. Gauss was thus led to in- troduce as modulus instead of p one of its imaginary factors ; an innovation which necessitated the construction of an arithmetical theory of complex imaginary numbers of the form a -J- bi." In a paper 2 communicated to the Royal Society of Gottingen, April 15, 1831, a report 3 of which is given in the " Gottingische Anzeigen" for April 23, 1831, Gauss continues his investigations in this subject, limiting himself still to the case where p is a posi- tive rational prime of the form 4^+ 1, a an integer not divis- ible by p. 1 'Works, Vol. I, p. 71. 2 Th. Res. Biq, Com. Sec, Works, Vol. II. §93- 3 Ibid., p. 169. 208 THE REALM k{%). He obtains by induction, but does not prove, theorems concern- ing the moduli of which certain special values of a (± 3, 5, — 7, — 11, 13, 17, — 19, — 23) are biquadratic residues, and those of which they are non-residues, but says in the above mentioned report: "Although all these special theorems can be discovered so easily by induction it appears nevertheless extremely difficult to find a general law for these forms, even if much that is common makes itself evident, and it is still more difficult to find proofs for these theorems. The methods used for the num- bers 2 and — 2 in the first paper can not be applied here, and if other similar methods such as that used in dealing with the first and third classes, could serve to solve the problem, they prove themselves, however, entirely unsuitable as foundations for complete proofs. One soon recognizes, therefore, that it is only by entirely new paths that one can penetrate into* this rich domain of the higher arithmetic. The author has already pointed out in the first paper that for this purpose a peculiar extension of the field of the higher arithmetic is indispensable, without, how- ever, explaining more fully wherein this consisted ; the design of the present paper is to make known the nature of this extension. It is simply that a true basis for the theory of the biquadratic residues is to be found only by making the field of the higher arithmetic, which usually covers only the real whole numbers, include also the imaginary ones, the latter being given full equal- ity of citizenship with the former. As soon as one has per- ceived the bearing of this principle, the theory appears in an entirely new light, and its results become surprisingly simple." This widening of the field of the higher arithmetic consists, then, in considering our integers to be all those numbers of the form a + bi, in which a and b are rational integers. The defini- tions of divisibility, prime number, etc., and the principal theo- rems relating to rational integers having been shown to have their analogues for the integers of this extended system, our realm k(i), as has been proved in the preceding pages, Gauss then develops briefly the theory of quadratic residues for the integers of this new system. Passing to the subject of biquadratic resi- THE REALM k(i). 20O, dues, he separates all integers not divisible by a given modulus into four classes, as follows : " If the modulus be a complex prime number, a + bi, where a is always assumed odd, b even, and k a complex number not divisible by a + bi, then, for the sake of brevity p being written for a 2 -+- b 2 , we have in all cases £i(p-D == i } i } — i } — i } mo d a -\- bi, and thereby all numbers not divisible by a 4- bi are separated into four classes, to which in the above order the biquadratic charac- ters 0,1,2,3 are ascribed." That is, the biquadratic character of an integer, k, with respect to a prime modulus, a -f- bi, is the exponent of the lowest power of i to which k i{p ~ x) is congruent, mod a + bi, where p = a 2 + fr 2 . " It will be observed that, when a + bi is a prime of the first degree, the fourfold classification of the real residues of a + bi which we thus obtain is identical with that obtained for p, =n[a-\-bi], in the real theory; for the numbers / and — f, being the roots of the congruence x 2 -f- 1 = o, mod p, satisfy the same congruence for either of the complex factors, a-\-bi, a — bi, of p, and are therefore congruent respectively to + i and — i, for one of these factors, and to — i and -f i for the other. 1 " Evidently the character o belongs to the biquadratic residues, the remaining ones, 1, 2, 3, to the biquadratic non-residues, to the character 2 corresponding quadratic residues, to the characters 1 and 3 on the other hand quadratic non-residues. " One recognizes at once that it is only necessary to determine this character for such values of k as are themselves complex primes, and here induction leads immediately to most simple re- sults. If, first of all, we put k = 1 + i, it is seen that the charac- ter of this number is always congruent to i( — a 2 -f 2ab — 3& 2 -f-i), mod 4, 1 See H. J. S. Smith : Works, p. 197. 14 2IO THE REALM k(i). and similar expressions are found for the cases k = i — i, — I -\-.% — i — i. " If, on the other hand, k be such a prime number c + di, that c is odd and d even, we can obtain by induction a reciprocity law quite analogous to the fundamental theorem for quadratic resi- dues ; this theorem can be expressed most simply in the following manner : " If c -\- d — i as well as a-\-b — i be divisible by 4 (to which case all others can be easily reduced), and the character of the number c -f- di zinth respect to the modulus a -f- bi be denoted by l lf that on the other hand of a-\- bi with respect to the modulus c -f- di by l 2 , then h — l 2 when one (or both) of the numbers d and b is divisible by 4; on the other hand l x — l 2 ± 2, when neither of the numbers d, b is divisible by 4. " These theorems contain in truth all the essentials of the theory of the biquadratic residues ; easy as it is to discover them by induction, it is most difficult to prove them rigorously, especially the second, the fundamental theorem of the biquadratic residues. On account of the great length of the present paper the author finds himself obliged to postpone to a third paper 1 the presenta- tion of a proof of the latter theorem, which has been in his pos- session for twenty years. On the other hand, the present paper contains the complete proof of the first theorem relating to the number 1 + h upon which are dependent the theorems relating to 1 — i, — I -(- i, — 1 — i. This proof will give some idea of the complexity of the subject." The above will be made plainer to the reader by the following brief resume. The integer a is said to be a biquadratic residue 1 Gauss never published his proof of this theorem, but soon after the theorem was published Jacobi succeeded in proving it, and communicated this proof to his pupils in his lectures at Konigsberg in the winter of 1836-37. He did not, however, publish his proof, and the first published proofs are by Eisenstein, who gave in all five. See Crelle, Vol. XXVIII, P- 53, P- 223, and Vol. XXX, p. 185; also H. J. S. Smith: Works, Vol. I, p. 78, and Bachmann: Die Lehre von der Kreisteilung, p. 168. THE REALM k(i). 211 or non-residue of a prime, *, a being prime to tt, according as the congruence x* = a, mod7r, is or is not solvable. From Th. 16 we have a w[7r]-1 = i f modTT, 3) and since, excluding the case ir= r + h 1 n [ 7r ] — I is always divis- ible by 4, we may write 3) in the form / nQ]-l \ / n [*•]-! \ / n[ir-]-l \ / n[n^-l \ [a 4 -i)\a 4 -i)\a 4 + 1) \a 4 +*j=o, modTr, each of the congruences »[tt]-1 t 4 "L*]-i t 4 w[tt3-1 4 , mod7r, a - ss — % which may be written in the common form n\_n]-l a 4 wmif, modTr, r = o, 1,2, 3, is seen to have exactly (n[Tr] — 1)/4 incongruent roots, and the integers of a reduced residue system, mod ir, fall into four classes according as they satisfy the first, second, third or fourth of these congruences. The integers of the first class are the biquadratic residues of it, for a 4 = 1, modTr, is the necessary and sufficient condition that a shall be a biquad- ratic residue of 7r. The integers of the first and third classes are together the quadratic residues of tt, for they are the roots of the congruence a 2 =1, modTr. 1 It is easily seen that every integer not divisible by 1 -f- * is a biquadratic residue of 1 + i. 212 THE REALM k(J). The integers of the second and fourth classes are together the quadratic non-residues of if, for they are the roots of the congruence n[tr]-l a 2 = — i , mod 7T. The exponent of the power of i for which the congruence n[ir]-l a 4 ■■ i r , mod r, r = o, i , 2, 3 is satisfied is called the biquadratic character of a with respect to it and this power of i is denoted by the symbol (o0r) 4 , so that we have always % ©.«~ mod The symbol (a/7r) 4 , which is due to H. J. S. Smith, seems preferable to (( ct / 7r )), which was adopted by Jacobi, as by a change of subscript it will serve for the theory of residues of other degrees. If now (a/ir) have the meaning previously assigned, we see easily that a)-©;- If we understand by the quadratic character of a, mod *", instead of 1 or — 1, the exponent of the lowest power of — 1 to which a is con- gruent, mod t, the notation for quadratic residues will be brought into accordance with that given above for biquadratic residues. The symbol (fit/*) 4 obeys the following laws: From a x = a 2 , mod r, it follows that ©.-(?).- If a x and a 2 be two integers, which may be equal, not divisible by w, then from and ( — ) =a 2 4 f mod 7r, (5),(5),-<°.«.> it follows that nOJ-l 4 , mod 7T, THE REALM k(i). 21 3 (¥).- (5), (5);, Since every integer a can be written in the form a = i r (i+i)* Pl p 2 -- Pn , where r = o, 1,2, 3; ^ = 0, or a positive integer; and p 1 ,p 2 ,---,pn are odd primary primes, we have W 4 = W 4 VflT j 4 Vtt j 4 w 4 * ' • w; and the determination of the value of [ - | is seen to be resolved Wi into the determination of the values of ( - | , I | and 1 — 1 where p is an odd primary prime. The following theorem gives a simple criterion for determining the value of (*/r)«: Theorem 21. If Tr = a-\-bi be an odd primary prime, then i has the biquadratic character o, 1, 2 or 3 with respect to the mod- ulus tt, according as we have a^=i, 7, 5 or 3, mod 8; that is, \a + bi) Since a + bi is an odd primary prime, we have either a = 4k + 1 ; b = 4k, or a = 4k + $', b = 4k+2, and hence \a -f bi) q2 + &2_l i 4 =/ 2 fe, when = 4^+1, = j 2&+3 , when a = 4^-(-3. But 2^ = oor 2, mod 4, according as k is even or odd ; that is, according as = 4^+1 = 1 or 5, mod 8 ; and 2k + 3 = 3 or 1, mod 4, according as k is even or odd ; that is, according as 214 THE REALM k{i). a = 4k + 3 = 3 or 7, mod 8. Hence ( — ^-.) ==1,*,— ior — i. according as a= 1, 7, 5, or 3, mod 8. The following table gives the biquadratic character of i with respect to each odd primary prime whose norm is less than 50. Biq. Char. Odd Primary Primes. O % i+4*, i—4*, —7- I — I + 2f, — I — 21, — 1+6*', — I - -6*. 2 — 3,5 + 4h 5— 4*. 3 3 + 2», 3 — 21, — 5 + 2f, — 5 — 2*. The following theorem gives the biquadratic character of 1 + * with respect to an odd primary prime modulus. Theorem 22. If a-\- bi be any odd primary prime (i±l\ _;~ b i*- 1 \a + bi)- 1 For the proof of this theorem see Gauss: Works, Vol. II, p. 135; Eisenstein: Crelle, Vols. 28 and 30; Bachmann: Die Lehre von der Kreisteilung, p. 181. The following table gives the biquadratic character of 1 + f with respect to each odd primary prime whose norm is less than 50. Biq. Char. Odd Primary Primes. O 3 — 2i,5 + 4i — 1— 61 I 1 — 4h — 5 + 2 i — 1 + 6* • 2 — 1 +2*, — 5 — 2f,5 — 4*, — 7. 3 — 1 — 2%, — 3, 3 + 2%, 1 + 4*. This theorem is easily seen to be the equivalent of Gauss' (p. 209), for although the modulus is here restricted to an odd primary THE REALM k(i). 21 5 prime, a-\-bi, while in Gauss' it can be either ±{a-\-bi), where a -f- bi is an odd primary prime, this makes no difference, since (i+i) _ ( ■**' ) We have only to show therefore that \{— a* + 2ab — 3 b*+i)^i(a— b — b 2 — i), mod 4 , 4) where a = 1, b = o, or a == — 1,6 = 2, mod 4. Putting a = 4a 1 -\-i,b = 4b 1 , or = 4^ — i,fr = 4& 1 + 2 in 4), we obtain in both cases (b 1 — a 1 )(2a 1 -{-2b 1 +i)=a 1 — b 1 , mod4; that is (»i + ^i + I )( fl i — &i)=o, mod 2, is a necessary and sufficient condition that 4) shall hold, and this condition is easily seen to be satisfied by all values of a x and b v The value of (a/7r) 4 is determined by means of the reciprocity law given by Gauss, which can be expressed most simply as follows : Theorem 23. The biquadratic characters of two odd primary primes of k(i) with respect to each other are the same or opposite according as one of the primes is =1, mod 4, or both are = j -\- 2i } mod 4. This can be expressed symbolically as follows : £).-<->---m in which ir and p are any two odd primary primes of k(i). Since we have from the last theorem (>)-(;):• (>)-(& and from this can easily deduce the quadratic reciprocity law as given in Th. 20. 2l6 THE REALM k (i) . The biquadratic character o is opposite to 2, and I to 3, this corresponding to l t = l 2 ± 2 in Gauss' theorem (p. 210) . His con- dition, that a + b — 1 and c + d — 1 shall both be divisible by 4, is evidently satisfied when the primes are primary. Furthermore, it is easily seen from the definition (p. 194) that every odd pri- mary prime is 5= 1 or 3 + 21, mod 4 ; and this is equivalent to Gauss' condition that b (or d) be divisible or not divisible by 4. Ex. 1. To determine the value of \5 + 4*A Resolving 1 + 31 into its primary prime factors, we have \S + 4*A \S + 4*/A5 + 4«/A 5 + 4* )i ByTh. 21 (- i —\ =i 3{b - 1)/2 = ?, \5 + 4*/4 and by Th. 22 (±+1 \ = i°. \5 + 4*7 Since — 1+2* and 5 -f- 41 are odd primary primes we have by Th. 23 and since 5 + 4* — (1 — 3*) (— 1 + 2 i)= — i; 1 that is, 5 -f 41 == — i, mod — 1 -f 2/, we have (_* + 4L \ / -»' \ / 1_Y = * V— I 4- 2* A \— I+2I/4 V— I+2t/ 4 Combining these results, we have \5 + 4*A that is, 1 + 31 is a biquadratic non-residue of 5 -f 41, or in other words the congruence x* == 1 -f- 3/, mod 5 -f 41 has no roots. We see also that 1 + 3* is a quadratic non-residue of 5 -f- 41. Ex. 2. To classify the odd primary primes of k(i) according to the biquadratic character of — 1 -f 2f with respect to each of them. Let it be any odd primary prime of k(i). 1 We select 1 — 31* as a* is chosen in Th. A. THE REALM k(i). 217 We have two cases to consider according as t^i or 3 + 21, mod 4. i. ''"^i, mod 4. Then V • /*~~ \— 1 + 2i7 4 — V— 1 4- 2iji \-i+ 21) \ ' \— 1 + 2i) t or( -* ) , V— 1 + 21/4 according as ■'as I. i — 1 or — i, mod — 1 + 21, 1, *', — 1, — t being a reduced residue system, mod — 1 + **> But fcfe) =«■ fcira).-* G^a)-'- (^),=<' 3 - Hence with respect to an odd primary prime, w t sp i, mod 4, — 1 + 2t has the biquadratic character o, 1, 2 or 3, according as we have *"==i, t, — 1, or — i, mod — 1 + 21. ii. * 35 3 + 2», mod 4. Since we have both » and — i+2i = 3-f 2#, mod 4, it follows that (-*+*) ( ' Y V * A V— 1+21/4 Hence with respect to an odd primary prime, t, ss3 -f 2 'j mod 4, — 1 + 21 has the biquadratic character o, I, 2 or 3, according as we have * as — 1, — *j 1 or i, mod — 1 + 2/. Combining these conditions we see that — 1+2* has with respect to an odd primary prime, *", the biquadratic character where ir=n( — 4 + &') + 1 or A*( — 4 + 80 + 3 + 21, 1 where t — fi( — 4 -|- &") + 1 -j- 4* or /*( — 4 + Si) + 3" — 2», 2 where 7r = /i(— 4-f 81) + 1— 4/ or /*(— 4 + &*) +3 + 6*, 3 where 7T — /u( — 4 + 8O — 3 or /*( — 4 + 8*) +7 + 21, M being any integer of k{i). Ex. 3. Determine whether the congruence x* = 9 + 7i, mod 5 + 4^ has roots. Ex. 4. Class the odd primary primes of k(i) according to the bi- quadratic character of 3 + i with respect to each of them. CHAPTER VI. The Realm &(V — 3)- § i. Numbers of &(V — 3). The number V — 3 is defined by the equation s2 +3== 0, I) which it satisfies. We can show exactly as in k(i) that all num- bers of &(V — 3) have the form a + &V — 3> where a and b are rational numbers. The other root, — V — 3> of 1) defines the realm k( — V — 3) conjugate to &(V — 3). These two realms are, however, evidently identical. The number a', — a — b\/ — 3, obtained by putting — V — 3 for V — 3 i n an y number a, = a-\-by/ — 3, of &(V — 3), is the conjugate of a; for example, 2+V — 3 an d 2 — V — 3 are conjugate numbers. y^> A rational number considered as a number of &(V — 3) is evi- ^jA dently its own conjugate. The product of any number, a, of &(V — 3) by its conjugate is called its norm, and is denoted by n[a] ; that is, n[a + b V=z\ = (a + b y- ^)(fl — &V=3)=a 2 +3& 2 We see that the norms of all numbers of k{ V — 3) are positive rational numbers. We can prove exactly as in k(i) that the norm of a product is equal to the product of the norms of its factors; that is, n[ap]=n[a]n[/3], where a and ft are any numbers of &V — 3- We observe, just as in k(i), that every number a, ==a -f- b\/ — 3, of &(V — 3) satisfies a rational equation of the second degree, that being the degree of the realm, and that this equation has for its remaining root the conjugate of a. The numbers of &(V — 3) fall then, as in k(i), into two classes, imprimitive and primitive, according as the above equation is 218 THE REALM k(\/ 3). 219 reducible or irreducible; that is, according as 6 = or=j=o. The imprimitive numbers are therefore the rational numbers, and the primitive numbers all the other numbers of the realm. It is evident that any primitive number of &(V — 3) can be taken to define the realm. This realm as well as the following ones will not be discussed as fully as k(i). Our desire is merely to bring out those points of difference from k(i) which necessitate some change in our conceptions, and to show that after these changes have been made and the unique factoriza- tion theorem proved for the integers of the realm, we can get as in k(i) a series of theorems analogous to those for rational integers. § 2. Integers of k ( V — 3). To determine what numbers of £(V — 3), in addition to the rational integers, are algebraic integers, we observe that as in k(i) the necessary and sufficient conditions that any number, a, = a-\-b^ — 3, of &(V — 3) shall be an integer are a + a! = a rational integer, and aa' = 3. rational integer. If we write a in the form a i + b i ^ -1 where a = a 1 /c 1 , and b = b 1 /c 1 , a x , b lt c t being integers with no common factor, these conditions become «t + ^i V — 3 ^-^iV-3 2a. . ■ . - 1 * + -3 ! = — ' = a rational integer, 1) c x c x v 1 l ) ( ) = 2 = a rational integer. 2) One at least of the three following cases must occur: i. c 1 =\=2or 1; ii. c x = 2\ iii. c 1 = i. i. The impossibility of i is proved as in k{i). ii. If q = 2, 2a 1 /c 1 can be an integer, and yet a x not contain the factor 2, a x 2 -f- 2>b x 2 being divisible by 2 2 when a x and & 2 are both odd. 220 THE REALM &(V 3). Hence c 1 = 2, in which case a ± and b x must both be odd; or ^=1. Hence every integer of &(V — 3) has the form J(° + ^V — 3) > where a and b are either both odd or both even, and all numbers of this form are integers. §3. Basis of MV^). 1 A basis of k( V — 3) w defined as in k(i). It will be observed that the integer V — 3 defining k ( V — 3 ) does not constitute with 1 a basis of the realm as i and 1 did in k(i) ; that is, there are integers of the realm that can not be represented in the form x -f- y V — 3, where x and 37 are rational integers. We shall see, however, that two integers of &(V — 3) can be found, which form a basis of the realm. For example, 1, £( — 1 + V — 3) 1S a basis of k{ V — 3) ; for let J( — 1 +V — 3)> which is seen to be an integer, be represented by p, and -J (a + ^V — 3) be any integer of k ( V — 3 ) . We shall show that J ( a + b V — 3 ) can be put in the form x + yp, where x and y are rational integers. _, a -f b v — 3 2x—y y put — - — 5 - = *+j, P — - T z+i v _ 3> which gives 2x — y = a, y = b, whence x = ^(a-\- b), y = b, a ,u £ a + bv — 3 a + b and therefore ~ = 1- bo 2 2 r ' where \{a-\-b) is a rational integer, since a and b are either both even or both odd. Every integer of &(V — 3) can be repre- sented therefore in the form x -\- yp, where x and y are rational integers; that is, 1, p is a basis of &(V — 3). Moreover, every number of the form x -f- yp can be put in the form J(a + by/ — 3), where a and b are both odd or both even, and hence is an integer of &(V — 3). For, supposing x and 3; known, and a and b un- known, we see from the above analysis that a and b will be either both odd or both even, according as y is odd or even. The sum, difference and product of any two integers of &(V — 3) is an integer of &(V — 3), for 1 See Chap. V, § 3. THE REALM k(\/ 3). 221 O + y P ) =b (*, + y lP ) = x±x 1 +(y± y t )p, and O + yp) (*i + y\p) —x*t + (*y, + *i:y)p + yjip 2 = **x — 3'3'i + O^i + x x y — vyjp, since p 2 -f p -f- i = o. §4. Conjugate and Norm of an Integer of &(V— ^3)- The conjugate of p is p'=|( — i — V — 3) =p 2 . Since P + p' = P + p 2 = — 1, and pp' = p 3 =i, p satisfies the equation ^r 2 + .r+i=o; that is, p and p 2 are the imaginary cube roots of unity ; therefore fc(V — 3) is called the realm of the cube roots of unity. If a, =a-{-bp, be any integer of &(V — 3)» its conjugate is a', = a-\-bp 2 . The conjugate of a -\- bp 2 is evidently a-\-bp*, = a + b P . Hence n[a] = (a-\- bp) (a + bp 2 ) = a* + ab(p + P *)+by = a 2 — ab + b 2 , which is seen to be a positive integer. For example «[3 + 2p]=9 — 6 + 4 = 7- §5. Discriminant of £(V — 3). 1 The discriminant of k{ V — 3) is the squared determinant 1 P I P 2 formed from a pair of basis numbers and their conjugates. Denoting it by d, we have d = -3- §6. Divisibility of Integers of fc(V — 3)- We define the divisibility of integers of fc(V — 3) exactly as we defined that of the integers of R and &(/), and all that fol- lowed from this definition in R and k(i) holds for k(\/ — 3). 1 See Chap. V, §§3, 4 ; the same remarks hold here. 222 THE REALM &(\/ 3). Ex. i. We see that 4 + 5P is divisible by 3 + 2p, since 4+5P= (3 + 2p)(2 + p) = 6 + 79 + 2P 2 = 4 + SP, since p 2 = — 1 — p. Ex. 2. On the other hand, 5 -f- 2/> is not divisible by 3 + p, since there exists no integer of k(y/ — 3) which when multiplied by 3 + P gives 5 + 2p; for let $ + 2p= (3 + p)(* + yp) = 3*+ (* + 3y)p + yp 2 1) = 3* — y + + 2y)p; thus x and 3/ must satisfy the equations $x — y = Sj x-\ r 2y = 2, which give x= 12/7, y = y 7 ; that is, 1) does not hold for integral values of x and y, and hence 5 + 2p is not divisible by 3 + p. Theorem i. If a be divisible by (3, then n[a] is divisible by n[jB]. For from a = fiy follows n[a]=n[/3]n[y] ; that is, n[a] is divisible by n[/3]. As was seen in k(t), the converse of this theorem is not in general true. A common divisor of two or more integers is defined as in R and k(i). § 7. Units of k ( V — 3) . Associated Integers. The units of &(V — 3) are defined, as in the case of the last two realms, as those integers of &(y — 3) that divide every integer of the realm. They therefore divide 1, and since every divisor of 1 is evidently a unit, the units may also be defined either as those integers of &(V — 3) whose reciprocals are also integers of k{ V — 3), or, since if c be a unit, n[e] must divide 1, as those integers of k(y — 3) whose norms are 1. To determine the units of k( V — 3) we l et «, =•*■ + 37>, be one of them, and put n[e]=x 2 — xy + y 2 =(x — iy)* + iy* = i, from which we see that y can have only the values o, 1 and — 1. THE REALM k(\/ 3). 223 y= o gives x 2 =i, x=i or — I, and hence €=: 1 or — 1 ; 3/= 1 gives x 2 — jr. 4- 1 as i, .r = o or I, and hence € = p , or 1 +p = — p 2 ; y== — 1 gives x 2 + x + I = I, ,r = o or — i, and hence € = — p, or — 1 — p = p 2 . Hence € can have any one of the six values ± i, ± p, ± p 2 , which are therefore the units of fc(V — 3)- As &(V — 3) contains the primitive sixth roots, i(i + V — 3) an< 3 i(i — V — 3), of 1, and hence the cube roots of 1, it might more properly be called the " realm of the sixth roots of unity." Taking 1, «, = l(i-f-V — 3), as a basis, we would have as the six units of the realm 1, w } « 2 , <a 3 — — 1, w 4 , w°, the six sixth roots of unity. The nomenclature used above is, however, the usual one, and hence has been adopted here. If two integers, a and /?, have no common divisor except the units, they are said to be prime to each other, or, excluding the units, to have no common divisor. The six integers, a, — a, pa, — pa, p 2 a, — p 2 a, obtained by mul- tiplying any integer, a, of &(V — 3) by the six units in turn, are called associated integers; for example, the six integers, 1 — 6p, — 1 -f- 6p, 6 + 7p, — 6 — jp, — 7 — p and 7 + p are associated. Any integer which is divisible by a is also divisible by — a, pa, — pa, p 2 a and — p 2 a. Hence in all questions of divisibility, asso- ciated integers are considered as identical; that is, two factors, one of which can be changed into the other by multiplication by a unit, are looked upon as the same. §8. Prime Numbers of &(V — 3). The definitions are identical with those in k(i). We can determine whether any integer of &(V — 3) is prime or composite by the method employed for the same problem in k(i), the process depending upon Th. 1. E x. 1. To determine whether 2 is a prime or composite number of kW-3). Put 2=(a + bp)(c + dp); then 4= ( a 2 — ab + b 2 ) (c 2 — cd + <P) , +^= 2 , 224 THE REALM &(V 3). whence we have either a 2 — ab + b 2 z=2, c 2 — cd -\- d 2 = 2, 1) or a 2 — ab-\-b 2 =i,c 2 — cd-{-d 2 = 4. 2) It is easily seen that 1) is impossible; for, if then I & I = 1 and similarly [ a | $x. 3) It is evident that no pair of values of a and b, which fulfil the condition 3), can satisfy 1). Hence 1) is impossible, and 2) is the only admissible case ; that is, a + bp is a unit. Therefore 2 is a prime number in fc(V~"3). Ex. 2. To determine whether 3 is a prime or composite number of KV-T). Put 3 = (a + &p) (c + dp) ; then 9 —(a 2 — ab + b 2 ) {c 2 — cd + cP). whence we have either a 2 — ab + b 2 = 3, <r — cd + d 2 =% 4) or a 2 — ab-\-b 2 = 1, c 2 — cd + d 2 = g. 5) Now, if a 2 — a& + 6*=i, a + ftp is a unit and hence 5) is not an actual factorization. If a 2 — ab + b 2 =(a— |Y 4 J then I & I g 2, and I a I i 2. 6) The possible values of fc which satisfy 6) are 0, ± I, ± 2. Considering them in turn we see that b = o, gives a 2 = 3, which is impossible, b = 1, gives a 2 — a -(-1=3, and hence a := — 1 or 2, 5 = — 1, gives a 2 + a -}- 1 = 3, and hence a = 1 or — 2, b = 2, gives a 2 — 2a + 4 = 3, and hence «=I, 6 = — 2, gives a~ -\- 2a + 4 = 3> and hence a = — 1, whence a + &P = — (1 — p), ± (2 -f- p) or ±(i+2p). Similarly c+rfp = ±(i— p), ±: (2 + p) or ±(i+2p), and we have 3= (1— p)(2 + p) = (— i+p)(_ 2 — p) = (i+ 2 p)(— I— 2p), the proper combinations of factors being selected by trial. All these factorizations are, however, considered as identical, since the factors in THE REALM &(V 3)- 225 each resolution are associated with the corresponding factors in the other resolutions. All these factors can easily be proved to be primes of k(V — 3), whence we see that 3 can be resolved into the product of two prime factors in £(V — 3), and that this resolution is unique. Moreover, all these factors are associates of 1 — p, and we have 3 = — P 2 (i— p) 2 . We could have seen directly from the equation denning the realm that 3 = _(V^3) 2 . Ex. 3. If we endeavor to resolve — 46 + 37P into two factors neither of which is a unit, we find that it can be done in seven essentially different ways, the factors in each product not being associated with the factors in any one of the other products. — 46 + 37P= (4 + 5P)(h + i8p) 7) = (-5 + 6p)(8 + p) 8) = (7 + 2p)(— 4 + 9P) 9) = (1— p)(— 43 — 3P) 10) == (i + 3P)(29 + 25P) 11) = (4 + 3P)(5 + 22p) 12) = (5 + 3P)(i + i7P) 13) We find, however, that none of these factors except 1 — p, I+3P, 4 + 3P> and 5 + 3P are prime numbers, and that we can resolve those which are not prime into prime factors in the following manner: 4 + 5P = (1 — P) (1 + 3P), 11 + i8p =: (4 + 3P) (5 + 3P) ; _5 + 6p=(i + 3 P)(4 + 3P), 8 + p=(i-p)(5 + 3P); 7 + 2p=z (i_p)( 4 -f 3P ), _ 4 _pc>p = (i + 3p)(5 + 3p); — 43 — 2>P — (i + 3P)(4 + 3P)(5 + 3P), 2Q + 25P= (1 _p) (44- 3 p) (5 + 3 p); 5 + 22P = (1 — p) (1 + 3 p) (5 + 3 p), i + i7P= (1 — P)(i+3P)(4 + 3P)- When these products are substituted in 7), 8), 9), 10), 11), 12), and 13) we obtain in each case — 46 + 37P= (1— P)(i+3P)(4 + 3P)(5 + 3P) \ that is, when — 46 + 37P is represented as a product of factors all of which are prime, the representation is unique. Having made these notions concerning the integers of £(V — 3) clear, we proceed to what will always be our first goal in the discussion of any realm; that is, to prove that every integer of &(V — 3) can be expressed in one and only one way as a product of prime numbers. 15 226 THE REALM &(V 3) §9. Unique Factorization Theorem for fc(V — 3). Theorem A. // a be any integer of k(\/ — 3), and p any integer of &(V — 3) different from 0, there exists an integer fi of k{y — 3) such that n[a — n(3] <n[p]. 1 Let a/fl = a + bp, where a = r + r lf b — s -f- s lf r and s being the rational integers nearest to a and b respectively, and hence We shall show that ft, = r + Sp, will fulfil the required condi- tions. Since a//3 — ix = r 1 +s lP , n [a/p — /*] = r t * — rrf, + s t * g |, whence n[a/p — fi] < 1, or multiplying by n[p], n[a — fxp] <n[p]. The proofs of the two remaining theorems which lead to the Unique Factorization Theorem and the proof of that theorem itself are now word for word identical with those in k(i) ; we shall therefore merely state these theorems : Theorem B. If a and p be any two integers of £(V — j) prime to each other, there exist two integers, £ and -q, of &(V — 3) such that a£ + p v =i. Theorem C. // the product of two integers, a and p, of &(V — 3) be divisible by a prime number, n, at least one of the integers is divisible by v. This theorem has, of course, the same corollaries as the corre- sponding one in k(i). Theorem i. Every integer of k(\/ — 3) can be represented in one and only one way as the product of prime numbers. 1 See note in k(i) which applies equally here. THE REALM &(V 3). 227 § 10. Classification of the Prime Numbers of &(y — 3). By a train of reasoning identical with that employed in k(i), it becomes evident that every prime, ir, of &(V — 3) is a divisor of one and only one rational prime. In order therefore to deter- mine all primes of &(V — 3), it is only necessary to find the divisors of all rational primes considered as integers of k( V — 3). Let 7T, =a-f- bp, be any prime of &(V — 3) and p the positive rational prime of which -n is a divisor. Then p = 7r a J 1) and hence p 2 = n[7r]n[a]. We have then two cases \n[a]=p, ' \n[a] = i. i. From n[?r] =inr' = p and 1), it follows that a = ir f . From n [it] = p we have a 2 — ab -\- b 2 = />, and hence since every positive rational prime, except 3, is of the form 3^ + 1 or $n — 1, we must have, excluding the case /> = 3, when p = w[tt], a 2 — ab-\-b 2 = 1, mod 3, or a 2 — ab-\-b 2 = — 1, mod 3. The first of these congruences has the solutions a= o; a=±i; a=i; a== — 1] fr„ ±I; b= o; 6^1; 6^_i}' raod 3. while the second has no solutions. Hence when a positive rational prime other than 3 is the product of two conjugate primes of k{\/ — 5), it has the form- pi + I. The case p = 3 is easily disposed of, for the equation p = a 2 — ab + b 2 = 3 is satisfied by a=i, b = — 1, which give 3=(i-p)(i-p 2 ); hence 3 is the product of two conjugate primes of &(y — 3). These factors of 3 are, however, associated, for i-P 2 = -P 2 (i-p), 228 THE REALM k(y/ 3). whence 3 = — p *-(i — P ) 2 , or 3=— (V— ~3) 2 ; that is, j, which is the only rational prime divisor of the discrimi- nant of &(V — 3), is associated with the square of a prime of ii. From n[a]=i it follows that a is a unit. Hence p is associated with the prime 71-; that is, p is a prime in &(V — 3). When p is of the form $n — 1, this case always occurs, for we have seen that in order to be factorable in &(V — 3)> a rational prime must either be 3 or of the form 311 -f- 1. We shall now show that every rational prime, p, of the form 3^+ 1 can be resolved into the product of two conjugate primes of £(V-"3)- The congruence x- = — 3, mod p, /> = 3 w+i, has roots; for (-3//0 = (-i/7>)(3//0, and if p = $k-\- 1, (— i//>)= 1, and (3/70 = (J/3), while, if £ = 46 + 3, (— i//0=— 1, and (3//0=— (J/3), and in both cases therefore (-3//>) = (/'/3) = (i/3)=i- Let a be a root ; then a 2 + 3 = o, mod £ ; that is, (a + V — 3)( a — V — 3)=o, mod />. Since 0+ V — 3 and a — V — 3 are integers of &(V — 3), p must, if a prime in &(V — 3), divide one of them; we must have, therefore, either g+V— 3==» ■ I — - 2) when u and v are either both odd or both even, or THE REALM &(\/ 3). 229 where u ± and v t are either both odd or both even. But 2) and 3) are, however, impossible, since ±pv=±i implies that v is even, and hence that p is a divisor of 1, which is impossible. Hence p is not a prime in &(V — 3), and, since the only way in which a rational prime is factorable in k( V — 3) is into two con- jugate primes, p is factorable in this manner. The primes of &(V — 3) may therefore be classified according to the rational primes of which they are factors as follows : 1) All positive rational primes of the form $n -\- 1 are factor- able in &(V — 3) into two conjugate primes, called primes of the first degree. 2) All positive rational primes of the form $n — 1 are primes in k(y — 5), called primes of the second degree. 3) The number 3 is associated with the square of a prime of the first degree. It can be easily proved as in the case of 2 in k(i), that 3 is the only rational prime which is associated with the square of a prime of the first degree in &(y — 3). We observe that in k(\/ — 3) as well as in k(i) the only rational primes which are associated with the squares of primes of the first degree are those which divide the discriminant of the realm. § 11. Factorization of a Rational Prime in &(V — 3) deter- mined by the value of (d/p). As in k(i), we can express the above results in a very con- venient manner by means of the discriminant/ d, of &(V — 3)- We have seen that, when p — yi-\-i, ( — $/p) = i; that is, (d/p) = i. When p = 3, d is divisible by p, which is expressed symbol- ically by (d/p) =0. Hence we can classify the rational primes according to their fac- torability in £(V — 3) as follows: When (-] = /, p = Tnr f ; that is, p is the product of two conjugate primes of the first degree. When |-J= — j, p = p; 23O THE REALM k{ V 3)- that is, p is a prime of the second degree. When (-\ = o, p = eir 2 ; that is, p is associated with the square of a prime of the first degree. The primes of &(V — 3) whose norms are less than 100 are 2, ! — P, 5, i + 3p> 4 + 3P i 5 + 3p» 5 + 6p, 7 + 3p» 7 + 6 p> 5 + 9p> 7 + 9/°, i + 9p» io + 3p, n + 3p. § 12. Cubic Residues. If a and m be rational integers and a be prime to m, a is said to be a cubic residue or non-residue of m according as the congruence x 3 = a, mod m, has or has not roots. As in the development of the theory of biquadratic residues, we saw that our field of operation must be not simply the rational integers but the integers of the realm k(i), of which the rational integers are a part, so in the theory of cubic residues we must take as our field of operation the integers of &(V — 3) ; that is, we must consider the congruence x 3 zz=a, mod /a, where a and /x are integers of k{ V — 3) and a prime to /x. Lack of space forbids even a brief discussion of this subject here but the reader should consult Bachmann: Die Lehre von der Kreistheilung, I4 te Vorlesung ; Jacobi : Works, Vol. 6, p. 223, and Eisenstein : Crelle, Vols. 27 and 28. CHAPTER VII. The Realm k(y/2). § i. Numbers of &(\/2). The number V 2 is defined by the equation x 2 — 2 = 0, which it satisfies. All numbers of k{ V 2 ) have the form a -\- by 2, where a and b are rational integers. The other root, — \^2,oi x- — 2=0 defines the realm k{ — V 2 )> conjugate to &(V 2 )- The two realms are, however, as in both the previous cases, identical. The conjugate of a, — a-\-by2, is a', = a — by 2. The product act! is called as before the norm of a and is denoted by n[a]. In n[a] = (a-\-by2)(a — by 2)= a 2 — 2b 2 we notice the first of a series of important differences between this realm and k(i) and k(y — 3)- The norm of a number of k(y2) is not, as heretofore, necessarily a positive rational number. It may be either a positive or negative rational number. This will easily be seen to be true of all quadratic realms defined by real numbers, while the norms of numbers of quadratic realms defined by imaginary numbers -are always positive. Realms of the first kind, as k( V 2 )> are called real realms; those of the second kind, as k(i) and &(V — 3), imaginary realms. We have evidently ; w [a/?] =n[a]n[j3], where a and /? are any numbers of &(\/2). . §2. Integers of k {^/.2 ). Writing all numbers of k( V 2 ) * n tne fo rm a = <h + biV2 where a lf b lf c 1 are rational integers, having no common factor, 231 232 THE REALM k(\/2). we can show exactly as in k(i) that a necessary and sufficient condition for a to be an integer is ^=1. Therefore all integers of k(\/2) have the form a-\-b\/2, where a and b are rational integers, and all numbers of this form are integers; that is, 1, V 2 is a basis of k{ V 2 )- §3. Discriminant of ^(V 2 ) • The discriminant of &(V 2 ) is the squared determinant 1.1 V~2\* |l -l/2| formed from a pair of basis numbers and their conjugates. Denoting it by d, we have d=8. § 4. Divisibility of Integers of k( V 2 )- The definition is identical with that given in R, k(i) and &(V — 3). For example, since 14 + 9V 2 = ( 2 + V 2 ) (5 + 2 V 2 ) 14 + 9V 2 is divisible by 2 + 2V 2 and 5 + 2\/2. On the other hand, since no integral values of x and y exist for which the equation 5 + 2 V 2 = ( 1 + 2 V 2 ) O + y V 2 ) is satisfied, 5 + V 2 is not divisible by 1 + 2 V 2 - § 5. Units of k ( V 2 ) • Associated Integers. The units of &(V 2 )> being those integers of k(\/2) which divide every integer of the realm, divide 1, and since all divisors of 1 are evidently units, they can be defined either as those integers of k{^2) whose norms are either 1 or — 1, or as those integers of k( -\f 2) whose reciprocals are also integers. Let €, = x -f. 3/y 2, be a unit of &(V 2 ) ; we have then either n[c] = i, or n[e] = — 1; that is i. x 2 — 2y 2 =i, or ii. x 2 — 2y 2 = — I. 1 1 The reader will recognize i and ii as special cases of Pell's Equation x 2 — Dy 2 =±i, a discussion of which will be found Chap. XIII, § 5. Here we shall treat the question from a different point of view. THE REALM &(V 2 )- 233 We can easily obtain many solutions of both i and ii, as, for example : \v=± i, y= o, c=± 1, x=± 7,,y=± 2, £=±3±2V2, x=±l 17, y= ± 12, €=± 17 ± l2\/2, x= ± 1, y= ± I, €=±l± V 2 , <*===£ 7, :y=± S,'.€=±:7.±5V3r, $• = ± 41, y= ±29, c= ± 41 ± 29 \/2. We shall now show that &(V 2 ) has indeed an infinite number of units, each of which can, however, be represented as a power of the unit I + \/2, multiplied by -f- 1 or — i. This unit I + V 2 is called the fundamental unit. Theorem i. All units of k{^/2) have the form ±{i + V 2 ) n > •where n is a positive or negative rational integer or o, and all numbers of this form are units of k(y/2). Let e=i +V 2 - We see that every positive power of c is a unit; for »[ c »] = (»[ € ])»=( — i)*=i or — 1. Hence e n is a unit. Moreover, since € " £ - w =i, e"" is a unit also; that is, all negative powers of c are units, Furthermore two different positive powers of c give always dif- ferent units; for, since c, = I +V 2 > is greater than i, the positive powers of c are all greater than I and continually increase. Hence no two are equal. Also, since € -* = i/<p t it is evident that c -1 is less than I and hence that the negative powers of c are all less than I and continually decrease. There- fore no two negative powers are equal, and no negative power is equal to any positive power. Hence every power of e is a unit of &(V 2), and two different powers give always different units. Therefore k( V 2 ) possesses the remarkable property of having an infinite number of units. We shall now show that the powers 234 THE REALM &(V2). of c multiplied by ± I are all the units of k( \/2) ; that is, if r) be any unit of k( V 2 )> it will be of the form where n is positive, negative or o. Let + &V 2 De an y un it °f &(V 2 )- Then a — b V 2 > — a + &V 2 an d — a — ^V 2 w ^ a l so be units of k( V 2 )- Denote that one of these four units which has both terms positive by -q x {b may be o), the remaining three will be — -q x , rj^ and — •>//. We shall show that where n is positive or o. Since Vi^ 1 , it follows that 7 j 1 = e n , Or € n <7 7l <€ TC+1 i) where n is a positive integer or o. We shall show that the latter case can never arise. Dividing i ) by c n , we have i < wfe < €, where ^/c* is a unit, since the quotient of two units is a unit. Let r) 1 /e n = x-\-y\/2. We have (x-\-y^2)(x — y^/2) = ±I, and hence, since x + yV 2 > x > it follows that \x — yV 2 l <i; that is — i < x — y V 2 < J - This combined with i <^ + yV 2 < i + V 2 2 ) gives o < 2x < 2 -f-V 2 > and hence, ^ being a rational integer, jr= i. But, if 4'= i, it- is evident that no rational integral value of y will satisfy 2), for positive values of y give i + yV^i+V 2 , and y = o, or a negative integer makes THE REALM k(^/2). 235 I +3>y2< 1. Hence i) is impossible, and we have and therefore — Vi == — c "5 and since 77^/ = ± 1, 77/ = ± i/c n = ± c n , and — q/== zp r n . Therefore, if 77 be any one of the four units 77^ — rj lf 77/, — 17/, that is any unit of &(V 2 )> we have where n is positive, negative or o. We can express all units of k{i) in the form i n , but obtain only the four different ones i,i, — 1, — i, since * 4 = l. Likewise we can express all units of &(V — 3) in the form ±p n , but obtain only the six different ones 1, — 1, p, — p, p-, — p 2 , since p 8 = i. Any two integers which differ only by a unit factor are said to be associated, and in all questions of divisibility are considered as identical. Thus, if a be a factor of /x, and n any positive or negative rational integer, the infinitely many integers ± e n a, that are associated with a, are also factors of p.. All these factors, however, are considered as the same. With this understanding, we shall find that the fact that k( V 2 ) nas an infinite number of units in no way interferes with our adopting definitions for prime and composite numbers of k{ \/2) identical with those used in the previous realms and proving the unique factorization theorem for the integers of &(\/2). § 6. Prime Numbers of k ( y/2 ) . The definitions are identical with those in the preceding realms and we can determine whether an integer is prime or composite by the methods employed in those realms. Ex. 1. To determine whether 13 + 12V 2 is prime or composite. Put 13 + i2V2= (a + b V2) {c + d^2) ; then —119= (a 2 — 2b 2 ) (c 2 — 2d 2 ). 236 THE REALM &(\/2). There are only four distinct cases to be considered J a~ — 2b- =17, .. j a 2 — 2fr 2 = — 17, | c 2 — 2d 3 = — 7. J c' 2 — 2d 2 = 7. ..... f a 2 — 2b~ = ± 119, 111 and iv. ^ \c 2 — 2d i =±i. Both iii and iv give c -f- rf\/ 2 a unii an d therefore need not be considered. As solutions of i we have a = ± 5, b=±2, c = ± 1, d=±2, which give I3 + I2\/2= (5 + 2V2)(l+2y2) = (_5_2V2)(— I— 2V2), the proper factors being selected by trial. Since neither of the integers S+V 2 » I + 2 y'2 is a unit, 13 + 12^2 is a composite number. Other solutions of i are a = ±7, b = ±4, c=±u, d = ±8, which give i3 + i2V^=(7-4V2)(n + 8V2) = (-7 + 4V 2 )(- II - 8 V2)- As solutions of ii we have a = ±i, b = ±3, c = ±5, d = ±3, which give 13 + 12^2 = (— 1+ 3^2) (5 + 3V 2 ) = ( I— Z\J~2)i— 5 — 3\/2)- We see, however, that all these factorizations can be derived from any particular one by multiplying the factors by suitable units, and hence are not different; that is, 7-4^/2 = e- 2 ( 5 + 2 y2), ii+8y2 = e 2 (i+2y2), — 1 + 3^2 = r*(s + 2^2), 5 + 3 V 2 = c C* + 2 V 2 )' where e= 1 +V 2 » anc * we nave * n general 13 + 12V2 = [± e«(5 + 2V2) ] [± r*(i + 2V2) ]• Ex. 2. Prove that \-\-2-\j2 is a prime. §7. Unique Factorization Theorem for fc(V 2 )- Theorem A. // a be any integer of k( ^ 2), and (3 any integer of &(V<?) different from 0, there exists an integer /a of k(^2) such that \n[a-rf]\<\n[p]\ 1 Let a/(3 = a-\-b-\/2, 1 See note to corresponding theorem in k(i) which applies equally here. THE REALM k(\^2). 237 where a = r-\-r 1 , b = s-\-s 1 , r and ^ being the rational integers nearest to a and b respectively, and hence We shall show that^u, = r -{- s\/2,wi\\ fulfil the required condi- tions. Since d/p—ti=r x + s t -s/2 t \n[a/p — fi] I = \r* r- 2S* \ ^ J, whence \n[a/P — /*] | < i, or, multiplying by \n[/3] |, |n[a — ^]| <|n[0]|. The proofs of the two theorems which lead to the unique factori- zation theorem and that of the unique factorization theorem itself are identical with those in k(i) and fc(V — 3) with the exception that the absolute value of the norm is substituted for the norm of an integer. This is evidently necessary whenever we make a comparison between two integers of k( V 2 ) similar to that made between rational integers when we say that one is greater in absolute value than the other. It is also necessary when we ex- press the result of an enumeration as a function of an integer of k(\ / 2). In k(i) and fc(V — 3) tne norms of all numbers were positive and hence were their own absolute values. The result of an enumeration being always a positive integer, the conception of the positive integer being indeed arrived at by considering it as representing the result of an enumeration, to express such a result as a function of an algebraic integer, a, we must have some function of a which is always a positive integer. Such a function is |w[a]|. Theorem B. // a and (3 be any two integers of k{^/2) prime to each other, there exist two integers, $ and r), of k( y<?) such that Theorem C. // the product of two integers, a and /?, of &(V<?) be divisible by a prime number, w, at least one of the integers is divisible by », Theorem 2. Every integer of k(\/2) can be represented in one and only one way as the product of prime numbers. 238 THE REALM k(\/2). § 8. Classification of the Prime Numbers of k ( y 2) . By a train of reasoning identical with that employed in the preceding realms, it becomes evident that every prime, n, of k(\/2) is a divisor of one and only one rationa l prime . In order there- fore to obtain all primes of &(V 2 ) it is only necessary to resolve all positive rational primes considered as integers of &(V 2 ) into their prime factors in that realm. Let 7r, = fl-f- frV 2. b e any prime of k(^/2) and p the positive rational prime of which it is a divisor. Then £= jtO^ I ) and hence / >2 -=E=^I^l^I^] • We have then two cases n[a\j==JK \n[a]= 1. i. From n[y] = Tnr' — p and 1) it follows that a = 7r'. Since every positive rational prime, except 2, is of one of the forms Sn ± 1, 8n ± 3, we must have (excluding the case p=2) 9 when £ = »|y], a 2 — 2& 2 = 1, mod 8, 2) or a 2 — 2& 2 = — 1, mod 8, 3) or a 2 — 2b 2 == 3, mod 8, 4) or a 2 — 2& 2 = — 3, mod 8. 5) The first of these congruences has the solutions as* ±ii ±1, ±3, ±3 , mod 8. b = ± 2, o, ±2, The second has the solutions 11-4:1. ± i, ±3 , ±3) odg & — =fci, ±3. ctii ±3J The last two have no solutions, for they give a 2 ^2& 2 ± 3, mod 8, and hence require that 2b 2 ± 3 shall be a quadratic residue of 8. But the only quadratic residues of 8 are J and 4, whence it follows THE REALM k(^/2). 239 that a necessary condition that 3) or 4) shall have a solution is 1 ==2b 2 ± 3, mod 8, or 4 = 2fr 2 ± 3, mod 8. All four of these congruences are easily seen to have no solu- tions, and 4) and 5) therefore have no solutions. Hence when a positive rational prime other than 2 is the prod- uct of two conjugate primes of k(\ / 2), it has the form 8n± 1. The case p = 2 must next be considered. The equation a 2 — 2b 2 = 2 is satisfied by a = ±2, & = ±i. Hence 2= (2 + yi) (2— yi) = (1 + V 2 ) (— I + V~2) ( V 2 ) 2 ; that is, 2, which is the only rational prime divisor of the dis- criminant of &(V 2) is associated with the square of a prime of ii. Since n[a\= 1, a is a unit. Hence p is associated with the prime, ?r; that is, p is a prime in k(\/2). When p is of the form 8n ± 3 this case always occurs, for we have seen that to be fac- torable in k{^2) a rational prime must either be 2 or of the form Sn ± 1. We shall now show that every rational prime, p, of the form 8n ± 1 can be resolved into the product of two conjugate primes of fc(Vl). The congruence x 2 = 2, mod p, p = 8n±i, has roots, for (2/p) = i when£ = 8w±i. Let a be a root ; then a 2 = 2, mod />; that is (fl+V 2 )( fl — V 2 )— °> m °d />• Since a +V2 and a — \/2 are integers of &(V2), />, if a prime of k(\/2), must divide either a +V 2 > or a — V 2 - This is, how- ever, impossible, for from a ±y2 = p(c + dy2), where c-\-d^/2 is an integer of &(V 2 )> ft would follow that pd=± 1, which is impossible, since £ and d are both rational integers and /> > 1. Hence £ is not a prime in &( V 2 ), and since the only way 240 THE REALM k(\/2). in which a rational prime can be factored in &(\/2) is into two conjugate prime factors, p is factorable in this manner. The primes of k{ V 2 ) ma y therefore be classified according to the rational primes of which they are factors as follows : i) All positive rational primes of the form 8n ± I are factor- able in £(y<?) into two conjugate primes, called primes of the first degree. 2) All positive rational primes of the form 8n ± 3 are primes in k{ y 2), called primes of the second degree. 3) The number 2 is associated zvith the square of a prime of the first degree in k{ ^ 2). It can be shown, as in the cases of 2 in k(i) and 3 in &( V — 3), that 2 is the only rational prime that is associated with the square of a prime of the first degree. We observe that 2 is the only rational prime divisor of the discriminant. § 10. Factorization of a Rational Prime in k{ y^) determined by the value of (d/p). As in k(i) and &(y — 3), the above results can be expressed in tabular form by means of the discriminant of k(~\/2). The formation of such a table will be left to the reader. § 11. Congruences in k{^/2). The unique factorization theorem having been proved for the integers of k(^2), a series of theorems analogous to those deduced in the case of the preceding realms can be shown to hold for the integers of k(\/2). Having defined the congruence of two integers of k(^/2) with respect to a modulus precisely as we defined that of two rational integers, we should find that there are, with respect to a given modulus fi, I 11 [fx] I classes of incongruent numbers, and can then deduce for the integers of k(\^2) Fermat's theorem and other theorems relating to congruences. § 12. The Diophantine Equations x 2 — 2y 2 = ±i, x 2 — 2y 2 =±p, and x 2 — 2y 2 =±m. 1 It is required to find the rational integral values of x and y 1 See Chap. XIII, § 5. THE REALM &(\/2). 24 I for which these equations are satisfied. Since the first member of each of the equations is the norm of x -f- y V 2, the problem reduces, in the light of what we have learned about the integers of &(V2), to that of finding an integer of k{ V 2 ) whose norm is the quantity constituting the second member of the equation. If a -f- & V 2 be such an integer, then x = ± a, y=±b, evidently satisfy the equation under consideration. We see also that, if any one of these equations has a single solution, it has an infinite number of solutions, for if x = a, y = b be a solution of the given equation, and (a + b\^2)e 2n = a 1 + b x y~2, where e= I +V 2 > an d n is any positive or negative integer or o, then since wlA + ^V 2 ] =n[(a-\-b\/2)e 2n ] =n[a-j-&\/2], x = a lt y = b x is also a solution of the given equation. Moreover, since no two powers of c are equal, the solutions obtained by giving n any two different values are different. Hence the solu- tions are infinite in number. We shall consider now each of the equations in detail. i. x 2 — 2y 2 =i, ii. x 2 — 2y 2 = — I. The necessary and sufficient condition that an integer of £(V 2 ) shall have the norm db I is that it shall be a unit. All units having the norm I are included in the form ±(i +V 2 ) 2n > and all having the norm — i in the form ±(i -|-V 2 ) 2n+ \ n being a positive or negative integer or o. Negative values of n repeat solutions given by positive values, since (i+V 2 )"" 1S the con- jugate of (i+V 2 ) n - Hence, if ±{i+V2) 2n = a + by2, x=±a, y = ±b, satisfy i, and if ±(i+V2) 2n+1 = a 1 + b 1 V~2 } x=±a lf y = ±b 1 , satisfy ii, and these are all the solutions of i and ii. 16 242 THE REALM k{^2). For example: ±(i+V2) 2 =± (3 + 2V2) gives (±3) 2 — 2(±2) 2 =i; that is x = ± 3 ; 3' = ± 2 are solutions of i ; while ± (i+V2) 3 =± (7 + 5V2) gives (±7) 2 — 2(±5) 2 = — 1; that is x=±7; y= ± 5 are solutions of ii. iii. x 2 — 2y 2 = p, iv. x 2 — 2y 2 = — p, where p is a positive rational prime. The necessary and sufficient condition that ± p should be the norm of an integer of &(\/2) is p = ± 1, mod 8, or p = 2. Hence iii and iv are solvable when and only when ^±1, mod 8, or p = 2. Let p=± 1, mod 8. If x = a } y = b be any solution of iii, all rntegers of the form (a ± b^2)e 2n = a 1 + b{\/~2 give solutions of iii, x= ±a ly y=±b 1 ; for n[(a ± b^2)e 2n ]=n[a ± b^/2] (— i) 2n = p, and all integers of the form ( a ± b y 2 ) € 2n+1 = a 2 -\-b 2 y2 give solutions of iv, x=±a 2 , y=±b 2 ; for n[(fl±^V2)€ 2 " +1 ] =n[a±: b^2](—i) 2n+1 = — p. These are easily seen to be all of the solutions of iii and iv. Ex. 1. To find all rational integral solutions of the equations x 2 — 2y 2 = 7, x 2 — 2y 2 — — 7. A solution of the first equation is Hence (3 ±V 2 ) ( J +V 2 ) 2nr gives all solutions of the first equation and (3±V 2 )(i +\/2) 2n+1 all solutions of the second. Thus for example (3 + V 2 ) (1 + \Z2) 2 = 13 + 9V2 : (3- V 2 ) (1 + V2)* = 5 + 3V2 (3 + V 2 ) ( 1 + \/2) = 5 + 4V2" (3 — V2) (1 + V 2 ") = 1 + 2V2" v. x 2 — 2y 2 = m, vi gives (±i 3 ) 2 -2(± 9 ) 2 = 7, gives (±5) 2 -2(± 3 ) 2 = 7, gives (±5) 2 -2(±4) 2 = - gives (±l) 2 -2(±2) 2 = - i. X 2 - - 2y 2 = — m, THE REALM &(V 2 )- 243 where m is a positive rational integer. Since m must be the norm of an integer of k(\/2), and hence must be factorable into two conjugate integers of k(y/2), the necessary and sufficient condi- tion that v and vi shall have solutions is that every rational prime factor, p, of m such that £ss ±. 3, mod 8, shall occur to an even power. If then m = Pip2 ' ' • Prq x 2ti q 2 2t2 ■ ■ ■ q* 2t ', where Pi,P 2 > " -,Pr = ± 1, mod 8, or = 2, and q lf q 2 , ...,^=±3, mod 8, we have m={ir x iz 2 ■ • • Trrq^q^ • • • g,*«) (77-/77-./ • ■ • wr'q % **q 2 u ■ ■ • £,**)» = (a + &V2) (a — &V 2 ) =a 2 — 2b 2 , and #== ± a, y = ± & are solutions of v. If we interchange any 77- in one factor of 1) with its conjugate, we shall obtain a different factorization of m unless «[tt] = 2, in which case the factoriza- tion is not different, since the factors of 2 are identical. Suppose this interchange of in and tt/, m[tt] =f=2, to have been made, giving m= (a x -\- &iV 2 ) ( a i — ^iV 2 ) =«i 2 — 2& x *. Then x = ± a ly y = ± b x are new solutions of v. Suppose that by these interchanges of one or more 7r's with their conjugates we obtain all possible different factorizations of m. Then by multi- plying a factor of each of these factorizations by the even powers of e in turn we obtain from each factorization an infinite number of solutions of v, and by multiplication by the odd powers of e in turn we obtain from each factorization an infinite number of solutions of vi, and these are all the solutions of v and vi. That is, if a-i + b ± V 2 , a 2 + & 2 V 2 , • • • , a t + b t V 2 be each a factor of a different one of the t factorizations of m, all solutions of v are given by (di± bi^2~)e 2n =Ci n f C?* n V 2 , whence x = ± r« , y = ± </» n , and all solutions of vi are given by (a* ± ^V 2 )e 2n+1 =^^ n + /iV2, 244 THE REALM k{\/2) whence x=±ei n , y=±fi n , where i= 1,2, •••, t, andn = o, 1, •• • . Ex. 2. To find all rational integral solutions of the equations x' — 2y = 1 19 2r = — 119. and x~ We have 119 = 7 . 17 = (3 + y 2") (3 — y 2) (5 + 2^2) (5 — 2y 2) = [(3 + V2)(5 + 2y2)][( 3 -y2)(5-2y2)] = (i9+ny2)(i9— ny2), or = [(3 + y2)(5_2y2~)][(3-y2)(5 + 2y2)] = (n_y 2 )(ii+y2"). Whence we see that (19 ± nV2)e2« an d (11 ± V2)e2» gj ve a u the solutions of the first equation, and (19 ± ii\/2)e2n+i anc i ( IX ± y 2 ) e 2n+i give all the solutions of the second. Thus, for example: (i9+ny2~)(i + y2) =4i + 3oy2" gives (±4i) 2 — 2(±3o) 2 = — 119, (19— ny2)(i + V2) — — 3 + 8V2" gives (± 3 ) 2 _ 2 (± 8) 2 =: — 119, (11 + y2~)(i + y2) = i3 + i2y^ gives (± n) 2 — 2(±i2) 2 = — 119, (11 — V2)(i + y^) =9 + 10V2" gives (±9)'- — 2(± io) 2 z= — 119. CHAPTER VIII. The Realm &(V — 5). § 1. Numbers of &(V— 5) - 1 The number V — 5 ls defined by the equation that it satisfies. All numbers of &(V — 5) have the form a -f- &V — 5, where a and b are rational numbers. The conjugate of a, = a-\-by — 5, is a', =a — by — 5; also 4a]=a 2 + 5& 2 , and n[a/3]=n[a]n[l3] §2. Integers of k ( V — 5). Writing all numbers of &(y — 5) in the form fli + K V^ a = — , where a lt b lf c x are rational integers, having no common factor, we can show exactly as in k(i) that a necessary and sufficient condition for a to be an integer is ^=1. Therefore all integers of fc(V — 5) nave the form a + b V — 5 where a and b are rational integers, and all numbers of this form are integers; that is, 1, V — 5 is a basis of &(V — 5). § 3. Discriminant of k( V — 5 ) • The discriminant of k(\/ — 5) is 1 1, v-s 2 i = — 20. \i, -V-S] §4. Divisibility of Integers of &(V — 5). The definition is identical w T ith that adopted heretofore. 1 Throughout this chapter see corresponding sections in k(i). 245 246 THE REALM &(V 5). Ex. I. We see that 1 -+- 5\/ — 5 is divisible by 2 -f- >/ — 5, since 1 + JSV^ = (2 + V- 5) (3 + V- 5) • Ex. 2. We see that 5 + 2 V — 5 is not divisible by 4 -f- V — 5, since 5 + 2V :=r 5= (4 + n/- 7 ?) O + W^) holds for no integral values of x and y. §5. Units of fe(V— 5). Associated Integers. The units of &(y — 5) are defined as were those of the pre- ceding realms, and as the norm of a number of &(V — 5) is always positive, the necessary and sufficient condition that c, = x -f- y V — 5, shall be a unit is n[e]=x* + 5y* = i, which gives y = o, jt = ± 1 . Hence 1 and — 1 are the units of k{ V — 5). The definition of associated integers and the conventions re- garding them are identical with those heretofore adopted ; that is, the integers a and — a, obtained by multiplying any integer a by the units 1 and — 1, are said to be associated, and in all questions of divisibility are considered identical. §6. Prime Numbers of &(V — 5). The definitions are identical with those in the preceding realms. Ex. 1. To determine whether 2 is a prime or composite number in Put 2 = U + y V=5) (« + *V— 5) J then 4 = (V + 5V 2 ) (w 2 + $v 2 ), fjr a + 5y 2 = 2 ( x 2 („* + 5 ^ = 2 ° r "- {«■ and hence Evidently i is impossible since x and y must be rational integers. From ii it follows that w + v V — 5 is a unit. Hence 2 is a prime in *(V=5). Ex. 2. To determine whether 1 -J- >/ — 5 is a prime or composite num- ber of KV- 7 ^)- Put 1 + \/=5= U + y \/^S) (« + ^V— 5) ; then 6= {x 2 + 5y 2 ) (w 2 -f 5zr), THE REALM fc(V 5)- 247 and hence i- < o .. or 11. I , \ u 2 + sir — 2 \ u 2 + sir = 1 from which it is evident as above that 1 + V — 5 is a prime in k ( >/ — 5) . We observe that we have in i-f- V — 5 the first instance of a prime number whose norm is not a power of a rational prime. We shall see later that a necessary and sufficient condition for the norms of all complex primes of any given quadratic realm to be rational primes is that the unique factorization theorem shall hold for the integers of the realm. From these two examples it is easily seen that 3 and 1 — V — 5 are also primes in k( V — 5). § 7. Failure of the Unique Factorization Theorem in k ( V — 5) • Introduction of the Ideal. We shall now attempt to establish the unique factorization theorem for the integers of k ( V — 5 ) and begin as in the fore- going realms by endeavoring to prove the following theorem : Theorem A. // a be any integer of &(V — 5), and f3 any integer of &(y — 5) different from o, there exists an integer p of k( V — 5) such that n[a — (jLp]<n[l3]. Let . a/p = a + by~=5, where a = r -\-r x , b = s + s lt r and s being the rational integers nearest to a and b, respectively, and hence Let fl = r-{-s\/~^5; then a/p — fJ L = r 1 -\-s 1 ^— ^5 , whence n[a//3 — fi] = r x 2 -f- 5^i 2 i % that is, when fi is determined as above, we may have in &(V — 5) n[a/fi — fx\ > 1 instead of < 1 as has been the case in the three previous realms. Hence the integer fx chosen as above will not necessarily satisfy the require- ments of the theorem. The method which has hitherto served us for the proof of this theorem therefore fails. 248 THE REALM k ( V 5). That this theorem actually does fail for some integers of fc(V — 5) i s evident from the following example. Let a = 3 and fi = I +y/^S, then We are to find an integer /x=, ^r + ^V — 5> sucn that «[a//3- M ] = (i-^) 2 + 5(-i-y) 2 <i, but this is impossible, for it is evident that for all rational integral values of y, including o, the term 5( — ^ — y) 2 is itself > i. The method of proof adopted for Theorem A is seen to be depen- dent upon the general form of the norm of a number r x -f- s^o, where I, w is a basis of the realm. We have thus in k(i), &(V — 3), k(y/2) and &(V — 5) respectively \n[r 1 + s 1 <»]\ = \r 1 2 + s 1 2 \, \rf— r^+sfl \r t 2 — 2S X % and \rt 2 + SsS\> and the method is successful if KU4> ft! si be a sufficient condition for Wft+A«ii < h which is seen to be the case in k(i), &(V — 3) and &(\/2) but not in &(V — 5). We can easily determine all quadratic realms in which this method of proof holds ; that is, those in which this way of select- ing 11 is always successful. Let k(-\Jm) be any quadratic realm, 1 v ' m being a root of the equation x 2 — m — 0, where m is a positive or negative rational integer containing no squared factor. When m = 2 or 3, mod 4, k(^m) has as a basis 1, \/m, and when m= i,mod 4, k(\^m) has as a basis 1, ( — 1 + y/ni)/2 (see chap. X, §6). In the first case, it is easily seen that 1 See Chap. X, § 1. THE REALM k(\/ 5). 249 and in the second, a/£ — n=r 1 + s 1 (— 1 + \/m)/2, which give respectively n [ r t -\- s x V m ] =r 1 2 — m s x 2 1 1 and » r 1 -\-s 1 — l=*t—Vi T~ s * Considering first the case m = 2 or 3, mod 4, we see that is a sufficient condition that \r x 2 — ms x 2 \ < 1 2) when m= — 1, 2, — 2 or 3; but when \m\ > 3, then 1) is evi- dently not a sufficient condition for 2). Considering now me I, mod 4, we see that 1 ) is a sufficient condition that m — 1 -^1 — — s x -\ < 1 4 when and only when m = — 3, 5 or 13. Hence Th. A and consequently the unique factorization theorem holds in the realms k(i), &(V — 2), k(yj2), &(V3)> £(V — 3>> £(V5)> ^(V^)- To these can be added &(V — 7)> f° r when M = — 7, which is = 1, mod 4, if to 1 ) we add the condition that, when simultaneously 1^1=1 and \s x \ =1, then the signs of r t and s % are to be chosen alike, we see that in all cases ki 2 — r 1 s 1 + 2s 1 -\ <i. Hence the theorem holds for &(V — 7). A further slight modification in the method of selecting /x will enable us to show that the theorem holds for k{ V — n)- It is easily seen that, if I'll < i/V5> kil < I/V5; then \r 2 — r^ + 3^1 < 1. 3) Moreover, if either \r t \ or {s^ or both = \, then we can choose the signs of t t and s x so that they are alike, and hence 3) holds. 2 50 THE REALM &(V 5). There remains the case 1/V54 fa I < */*« I/V5 i kll < : / 2 * i. If r x and s t have like signs 3) evidently holds. ii. If r t and s t have opposite signs, for r ± we can put r 2 =^+1 or r x — 1, according as r x is negative or positive, hav- ing then VS and r 2 of the same sign as s 19 in which case ^2 2 -Vi + 3^i 2 < 1. Hence Th. A holds for &(y^Ti). It can be easily seen that the original method of selection, even when modified as above, will give a suitable value of /x in no imaginary quadratic realms other than those enumerated above, and it is furthermore evident that these are the only imaginary quadratic realms in which the theorem holds. It will be observed, as has been said in k{i), that Th. A is equivalent to saying that in a given realm we can find for any integer /? a complete residue system such that the norms of all the integers composing it are less in absolute value than n[(3]. This point of view is illustrated graphically in Chap. V, § 8. It must be carefully noticed, however, that although Th. A is a sufficient condition for the validity of the unique factorization theorem, it is not a necessary condition, as will be shown later. The proof of the theorem : Theorem B. If a and (3 be any two integers of &(V — 5)> prime to each other, there exist two integers, £ and q, of &(V — 5) such that has been heretofore based upon Theorem A, which has been seen not to hold for &(V — 5). This, however, would not, of course, justify the assumption that Th. B does not hold for &(V — 5), Th. A being a sufficient, but, as we shall see later, not a necessary, condition for the validity of Th. B. Nevertheless, the following THE REALM k(\/ 5). 25 I simple example will show that Th. B does not hold in general for the integers of &(V — 5). Let a = 3, /?=i+V =r 5- We have already seen (§6) that 3 and 1 -j-V — 5 are prime numbers ; moreover, they are not associates. Therefore they are prime to each other. We shall show that it is impossible to select two integers, $, =x-\-y\/ — 5, and 77, =f*-|-yy — 5, such that a$ + pr,= i 4) H 3(* + 3'V— 5) + (i+V— 5)(« + W— 5) = i, then 3.1- -\- a — $v=l t and hence 3-r — 33' — Ov = 1 , which is impossible since the first member only is divisible by 3. Therefore £ and rj can not be found so as to satisfy 4) and the theorem does not in general hold for the integers of k{ V — 5). We shall see later (p. 316) that the theorem: Theorem C. // the product of two integers, a and (3 of fc(V — 5) be divisible by a prime number, w, at least one of the integers is divisible by r, which is a necessary as well as sufficient condition for the unique factorization theorem, requires Th. B as a necessary condition for its validity. The following example will suffice to show that Th. C and the unique factorization theorem do not hold for the integers of &(V — 5). We have 6 = 2- 3 =(i+V : ^5)(i-V-l), and we have shown (§6) that 2, 3, 1 +V — 5 and 1 — V — 5 are prime numbers in &(V — 5). Moreover, the factors of one product are not associated with the factors of the other. There- fore 6 is represented in tzvo zvays as the product of prime factors. That this is not merely a peculiarity of 6 is seen from 21=3.7 =(i+ 2 V :zr 5)(i — 2 V^5), 9 = 3 2 =(2+ V=5)(2— V^5), and 49= 7 2 = ( 2 _|_ 3 y_^) (2 — 3^/^5), the factors in the above products being easily proved to be primes of &(y — 5). 2 52 THE REALM fc(V 5)- Moreover, that this failure of the unique factorization law does not occur in &(V — 5) alone may be shown by an examination of the realms k(\/~— -23) and fe(V — 89), in which we have respectively 27 = 38 = (2 +V— 2 3)(2— V— 23), and i2 5 = 5 3 =(6+V-89)(6-V— 89): 3,2+ V — 23 and 2 — V — 23 being prime numbers of fc ( V — 2 3 ) > and 5, 6+V— -89 and 6 — V — 89 being prime numbers of fc(V-89). It can now be made clear why we could not define the greatest com- mon divisor of two integers, a and /3, i. As the common divisor, 8, of greatest norm. ii. As the common divisor, 8, such that a/5 and j8/S are prime to each other. If a =r (1 -y-^) (1 +y=5) 2 = 6(1 +V-D. and /3 = 2(1-^/^5), then the common divisors of a and j8 other than the units are 2 and 1 — y/ — 5. Of these 1 — yj — 5 has the greater norm, 6, but 1 — y/ — 5 is not divisible by 2. Hence 8 so determined has not the important property of being divisible by every common divisor of the two integers. Considering the definition ii we see that there are two values of 8, 2 and 1 — V — 5> which satisfy it, for a/2 and P/2 are prime to each other, and and ■• ■ ■■ have the same property. Hence the defini- 1— V— 5 1— V— 5 tion ii, in addition to not determining 8 so that it is divisible by every common divisor of a and ft, does not even determine it uniquely. It is interesting to see, however, that, if we can find in any realm a common divisor, 8, of two integers a and P, such that every common divisor of a and /3 divides 8, then 8 will satisfy both the requirements i and ii ; for, considering i, if 8 X be a common divisor of a and /3 it divides 8; that is, 8 = «!/*, whence w[8] = rajA] . w[>], and therefore either |w[8i] | < |n[3] | or |»[»i] I = !»[«] I- In the latter case n|>] = ± 1, and hence a* is a unit ; that is 8 and 8 2 are associated. Hence 8 satisfies i. Considering ii, we have a = 8<3i and § = 8/3^ Now if at and A be not prime but have a common divisor, 6 1} then 8 would THE REALM &(\/ 5). 253 not be divisible by every common divisor of a and j3, for it would not be divisible by Sd t . We now ask whether it would be possible to deduce for the integers of &(V — 5), without the use of the unique factorization theorem, the series of theorems which have flowed from it for the integers of R, k(i), &(y — 3) and &(\/2). It is easily seen that in general these theorems do not hold in k ( V — 5 ) . For example, the analogue for k ( V — 5 ) of Fermat's theorem would be: // 7r be any prime of k( V — 5) and a any integer not divisible by 7r, then a »[7T]-l I EE= O, mod 7T, and indeed, if . v tt = 2 and a=i-\-2y/ — 5, 2 being a prime and 1 + 2 V — 5 evidently not divisible by 2, we have (i + 2V :ir 5) n[2] - 1 — i = (r + 2V— 5) 3 — 1 = — 60 — 34V— \5 = o, mod 2; that is,' the theorem holds in this case . But if 7T = 2 and a=i+V — 5> we see that, although 2 and 1 +V — 5 satisfy the requirements 2 a prime and 1 -f V — 5 not divisible by 2, (1 + V — 5)"™- 1 — 1 = (1 + V-^5) 3 — 1 = — 15— 2V— l^o, mod 2. The cause of this peculiar difference in the behavior of i+ 2 V — 5 an d 1 +V — 5 towards 2 in this relation will be made clear later (p. 379). Our next thought is can we by the introduction of a new conception of numbers reestablish the unique factorization law for the integers of fc(V — 5) when the factorization is expressed in terms of these new numbers. The introduction of the so-called ideal 1 numbers accomplish this, the primes of fc(V — 5) being in this widened number domain no longer in general looked upon as primes, but as being factorable 1 The term ideal number is used here in a general sense and is not to be taken to refer particularly to the ideal numbers of KTImmer. 254 THE REALM &(V 5). in terms of these ideal numbers. When this factorization has been performed we shall find that every integer of fc(V — 5) can be represented in one and only one way as the product of prime ideal numbers. The following considerations will make clearer their nature, and the ideas which have led to their conception. Let us con- sider the narrowed number domain composed of all positive rational integers congruent to 1, mod 5; that is, 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, etc. 5) Our definitions of divisibility and prime number being the same as before, we see that, when our operations are confined to num- bers of this domain, the unique factorization law does not in general hold ; for example, 336= 6-56 =16. 21, 1806 = 21-86 = 6-301, 1296= 6 4 =i6-8i, and 6, 16, 21, 56, 81, 86 and 301 are easily seen by multiplication of the numbers 5) to be prime in this domain. The cause of this failure of the unique factorization law is at once seen to lie in the absence of the remaining positive integers. As we suppose these integers to be unknown to us and in fact to have no real existence, we ask by what train of reasoning are we led from the requirements of the task to be accomplished, that is, the reestab- lishment of the unique factorization law, to the introduction of these missing integers, or rather the introduction of symbols which have their properties so far as the task in hand is concerned. Consider 336 = 6-56=16-21. Since 6 is not contained in either 16 or 21, although the product 16-21 is divisible by 6, we suppose 6 to be the product of two factors one of. which is contained in 16, the other in 21, and denote these factors by (6, 16) and (6, 21), respectively. The factor (6, 16) plays the same role with respect to 6 and 16 in all questions of divisibility in which these new numbers are used that the greatest common divisor of two integers plays with re- THE REALM &(V 5). 255 spect to these integers when only the original numbers of the domain are involved. We can therefore in this sense consider (6, 16) as the greatest common divisor of 6 and 16. Likewise we consider (6, 21) as the greatest common divisor of 6 and 21, and we write 6=(6,i6)(6,2i), denoting by this equation that 6 and the product (6, 16) (6, 21) in all questions of divisibility play the same role; that is, every integer that is divisible by 6 is divisible by (6, 16) (6, 21), and conversely. This convention is evidently justified by the fact that in reality (6, 16) is 2 and (6, 21) is 3. Similarly we have 56= (56, 16) (56, 21), 16 = (16, 6) (16, 56), 2I = (2I,- 6) (21, 56), and hence 336 = 6.56= (6, 16) (6, 21) (56, 16) (56, 21) = 16.21 = (16, 6) (16, 56) (21, 6) (21, 56), and the factorization is seen to be the same, the change of order of the numbers in the parenthesis having no effect on the symbol; that is, (6, 16) = (16, 6), etc. We have now seen that the failure of the unique factorization law in a certain number domain can be remedied by the introduc- tion of a new kind of number each of which is defined by a pair of integers of the domain and may be looked upon as the greatest common divisor of these integers. These numbers might be called the ideal numbers of the domain, and although the fact that the numbers of this domain do not form a realm prevents our expanding their conception and definition to the extent that we shall now develop those of the ideal numbers of &(V — 5), still we shall find that the same conception will enable us to reestablish the unique factorization law in this realm. We shall not, however, conceive of these new numbers, which we are about to introduce into k(\/ — 5), simply as being each the greatest common divisor of a pair of integers of k{ V — 5) and as defined by these integers, but as being each the greatest common divisor 256 THE REALM k(\/ 5). of an infinite system of integers of &(V — 5) and as defined by any finite number of these integers such that all other integers of the system are linear combinations of these with coefficients which are any integers of the realm. These numbers we shall call the ideal numbers, or briefly the ideals of &(V — 5). To make this clearer, consider the equation 2-3=(i+v zr 5;)(i— v^s). Since 2 divides neither (1 +V — 5) nor (1 — V — 5), although it divides their product, we must, to reestablish the unique factori- zation law, consider 2 as the product of two ideal factors, a and h, 1 which divide 1 +V — 5 and 1 — V — 5 respectively, the quo- tients being supposed, of course, to be ideal numbers also. We can denote a and h by the symbols (2, 1 +V — 5) an d ( 2 » 1 — V5) respectively. If now a be considered to bear the rela- tion of greatest common divisor to 2 and 1 -f-V — 5> it will bear this relation to the entire system of integers, which are linear combinations of 2 and 1 -f-V— 5; that is, those of the form 2a -\- (1 +V — 5)/?, where a and j3 are any integers of the realm. Conversely, if a be considered to bear this relation to the entire system, it will bear it to 2 and 1 +V — 5- We consider then a to be determined not by 2 and 1 -f-V — 5 alone but by this entire system of integers, and by a natural transition say now that a is this system of integers. We write therefore understanding by this symbol the entire system of integers which are linear combinations of 2 and 1 +V — 5> w ^h coefficients which are any integers of the realm. In order to define a, it is therefore sufficient to give any set of integers such that all linear combinations, with coefficients as above, exactly constitute the above system. Hence we can introduce into the symbol defining a any integer that is a linear combination of those already there, and can omit any integer that is a linear combination of those remaining ; thus : 1 Ideals will be denoted by German letters. THE REALM &(\/ 5). 257 a=(2, i+v— 1) = (2, i+V— "5, 2 + 2^— ~5, 3 + 3 V^5) = {2, 2 + 2V =7 5, 3 + 3V— "5)- The object of the preceding discussion, that has been by no means rigorous, has been first to show the necessity for the intro- duction of ideal numbers, and second to acquaint the reader in some degree with the ideas which have led to their conception and which induce us to adopt the definition which we shall now give. The justification of this definition will be found in the fact that, after we have defined what is meant by the equality of two ideals and what is meant by their product, we shall see that, when the integers of &(V — 5) are resolved into their ideal factors, the unique factorization law will be once more found to hold. More- over, we shall see that the behavior of an ideal towards the integers of the system constituting it is such as to warrant our original conception of an ideal as the greatest common divisor of this system. § 8. Definition of an Ideal of jfe(y— 5). An ideal of k(y — 5) is an infinite system of integers composed of all linear combinations of any finite number of integers, a x ,a 2 , '-,a n , the coefficients being any integers of the realm. 1 The integers a x ,a 2 , --.a n are said to define the ideal and the integers of the infinite system of integers constituting the ideal are called the numbers of the ideal. If an ideal a be defined bv the integers a^a^, •••,<*» we write a= (a lt a 2 , •••,a»), understanding thereby the infinite system of integers of the form SA+$AH h&flt* 1) where £ x , £ 2 , ••-,£« are any integers of the realm. We shall call (a lf a 2 , • • •, a„) the symbol of the ideal of a. ^^The general definition of an ideal of any quadratic realm (Chap. XII, §1) seems at first sight broader than this definition, but as it is shown that all the numbers of any ideal are linear combinations of a finite num- ber of them, the definitions are equivalent. 17 258 THE REALM fe(V 5)- If y be one of the integers included in i) ; that is, if y == kfit + X 2 a 2 -f- • • • -L. X n CLn, where A^ A 2 , • • •, X n are integers of the realm, we have a=(a 1 ,a 2 , • • •, a») = (a lf a 2 , •■••,a»,y), 2) for the infinite system of integers of the form lyA'H- ^2 a 2 + h yna n + ^ w+1 y, 3) where t) x ,-q 2 , ■•■,r} n+1 are any integers of the realm, is the same as the system 1), since putting the value of y in 3), we have ill + Vn+iK)^i + (V2 + yn +1 X 2 )a 2 -j h(Vn + r)n + lXn)a n , a system that evidently coincides with 1). It is evident then from 2) that we may, without changing an ideal, introduce into its symbol any integer which is a linear combination of those already there, the coefficients being integers of the realm, and may omit from the symbol any integer which is a linear combi- nation of those remaining. § 9. Equality of Ideals. Two ideals, a= (a lt a 2 , • -,a m ) and B = (fi lf yG 2 , • • •, £»), are equal when the two infinite systems of integers that constitute these ideals are the same. The necessary and sufficient condition for this is that every number, a x ,a 2 , •■•,a m , defining a shall be linear in the numbers, 1 ft,/? 2 , •••,£»», defining B, and that every /? shall be linear in the a's ; that is, it is necessary and sufficient that we shall be able to introduce the numbers a x ,a 2 , •••,a« into the symbol of B, and the numbers /3 X , (3 2 , '-,(3 n into the symbol of a; in other words, zve must be able to reduce the symbol of either one of the ideals to that of the other. Ex. 1. To prove that the two ideals a = (2, 1 + V — 5), and h= (2, 1 — V — 5)> are equal. We have (2, i + V :i: 5) = (2, i + V 1 ^ 1 — V— ~5), since 1 — v^ = 2(— Vj^S) + (1 + V-HS) I and (2, 1 + y/^5, 1 — y/^l) = (2, 1 — V^S) 5 since I + V^-S — (y/—$2^ (t — y/ZZJ). 1 When we say that cu is linear in ft, ft, •••, /S w we shall understand that a% = lift -f | 2 ft H h^»> where &, l 2 , ■••, In are integers of the realm. THE REALM fc(V 5). 259 Having reduced the symbol of a to that of B, the two ideals are seen to be the same. Ex. 2. To prove that the two ideals a= (3, 1 +V — 5), and h= (3, 1 — V — 5), are unequal. If we can show that any number, as 1 -f-V — 5, of a is not a number of b, the two ideals will evidently be unequal. If 1 -f- V — 5 be a number of B, then two integers, x + yV — 5, u-\-v\J — 5, of &(V — 5) must exist such that 1 + V^S = U + y V — 1)3 + O + vy/^S) (1 — V — 5), and hence 1 = 3* -f- « + 5^, 1 = 3y + » — u, whence by addition 2 = $x -f- 3y -f- 6v, an equation between rational integers that is impossible, since 3 is a divisor of the right hand member but not of the left hand member. Hence the required integers do not exist, and 1 + V — 5 is therefore not a number of the ideal b. The ideals are therefore unequal. Ex. 3. To prove that the two ideals a =(2, i + V — 5)> and b=(4, 2-\-2\J — 5), are unequal. Although, as is easily seen, the numbers denning the second ideal may be introduced into the symbol of the first ideal, we cannot introduce the number 2 of the first ideal into the symbol of the second ; that is, we can- not find two integers, x + yyj — 5, u -f v\J — 5, such that 2 = O + y V Zr 5)4 + (« -f v^/=5) (2 + 2yJ~—$), for from this equation it would follow that 2 = 4X + 2M — IOZ/, o == 4y -f- 2M -f- 2V, whence by subtraction 2 = 4-r — 4y — \2v, an equation in rational integers that is impossible, since 4 is a divisor of the second member but not of the first member. The two ideals are therefore unequal. Ex. 4. Show that (2, 1 +V=5) =j= (3, 1 +V^5). Ex. 5. Show that (29, 32 — 27V z:r 5) = (3 + 2V =r 5)- Ex. 6. Show that (49, 21 — /V— 5, 2I + 7^J^ r S, 14) == (7). Ex. 7. Show that (3 — V^l. 1 + 2y/^J) = {7, 3 — V :Zr 5)- 260 THE REALM k(\/ 5). § 10. Principal and Non-Principal Ideals. If among the numbers of an ideal, a, there exist a number, a, such that all numbers of the ideal are multiples of a, then a is said to be a principal ideal, and we have o=(a). If such a number does not exist, a is said to be a non-principal ideal. The necessary and sufficient condition for a to be a prin- cipal ideal is evidently that we shall be able to introduce into the symbol of a a number a such that all the numbers defining a are multiples of a. If such a number cannot be introduced, a is a non-principal ideal. Let us consider a few ideals with a view to determining whether they are principal or non-principal ideals. i. (7)^(2 + V^5), (6, 8, 2 + 6 V=S), (3, 3V=S), (3, V— 5), (5, V— 5)- ii. (2, i+V— 5), (3> i+V— 5), (3, 1— V— _5)- Considering those of the set i, (7) and (2 +V — 5) are seen at once from the definition to be principal ideals ; also (6, 8, 2 + 6V— 1) = (6, 8, 2 + 6V— 1,2) = (2), (3, 3V— ~5) = (3), (3, V zr 5)= 5: (3i V— "5,— 5) = (3 J VF-S>.— 5- 0=<l)> (5, V-1) = (V-1). Hence all ideals of the first set are principal ideals. Consider now the ideals of the set ii. If (2, 1 +V — 5) be a principal ideal, then there must exist a number, a, of the ideal such that 2 and 1 +V — 5 are both multiples of a. The numbers 2 and i"+V — 5> being primes in &(V — 5) and not associated, have as their only common divisors zb I. Hence a must be 1 or — 1. Since, if 1 be a number of the ideal, — 1 is also one of its numbers and vice versa, it is sufficient to see whether we can find two integers x + y V — 5 and u + ^ V — 5, such that 1 = 2<> + 3'V— 5) + (1 +V— 5) O + ^V— 5). We have from 1 ) 1. = 2x + u — 5 V > o = 23/ -j- v + u, THE REALM &(\/ 5). 26 I which give by subtraction I sa 2X — 23/ — 6v, an equation in rational integers that is impossible, since the second number only is divisible by 2. Hence 1 is not a number of the ideal (2, 1 -|~V — 5), and this ideal is therefore a non-principal ideal. Ex. 1. Show in like manner that (3, i + V — 5) an d (3» J — V — 5) are non-principal ideals. Ex.2. Show that (7, i + 2 n/—"S) and (7, 1 — 2y/ — 5) are non- principal ideals. Ex. 3. Show that (21, 9 + 3 V — 5, — 2 + 4V — 5) is a principal ideal. Had we introduced the conception of the ideal in the realms k(i), k( V — 3) and k(\/2), we should have seen that in all these realms every ideal is a principal ideal, for if a, = (a lf a 2t •••,a„), be an ideal, defined as above, of any one of these realms, then, since the unique factorization law holds in all these realms, we could in every case find integers £ r ,€ 2 , -',£n such that &0, + i 2 a 2 -\ f- inPtn = 5, where 8 is the greatest common divisor of a x ,a 2 , "',On* Hence we have a= (a lt a 2 , ••-,a n ,8) = (8), a principal ideal. On the other hand, we have seen (Th. B) that it is not always possible in &(V — 5) to find the integers £ lf £ 2 , ••-,£»; hence the fact that not all ideals of &(V — 5) are principal ideals. § 11. Multiplication of Ideals. By the product of two ideals a,= (a lt a 2f •••,a w ), and 6,= (&,&, •••,£«), we understand the ideal defined by all possible products of a num- ber defining a by a number defining h; that is, ab = (ajlv a x p 2 , • "lOifin, - -' t a m p lf • • •, aW?„). In other words, the product of a and b is the ideal whose numbers are all possible products of a number of a by a number of B, together with all linear combinations of these products. It is evi- dent from the above definition that the commutative and asso- 262 THE REALM k(\/ 5). ciated laws hold in the multiplication of ideals; that is, ah = ha and ab-c = a-hc. Ex. 1. (3, 1 + V—5)(3, 1 — V— 5) = (9, 3 — 3V— 5> 3 + 3\/- r 5' 6 >- = (9, 3 — 3^—^3 + 3^—5, 6, 3), = (3). Ex. 2. (2, 1 + V^S)"= (2, 1 + ^trj) ( 2 , j + V ZT5), =5 (4, 2 + 2^=5, —4 + 2^/^5), = (4, 2 + 2y— 1, _ 4 + 2 y^5, 2), since 2 + 2\/ — 5 — ( — 4 + 2\/ — 5) — 4 = 2. Hence, since all numbers in the symbol are multiples of 2, which is a number of the symbol, (2, i + V-5) 2 =(2). Ex. 3. (2, i+y^5)( 3 , !+y~5) — (6, 2 + 2^=5, 3 + 3\/— 5^ — 4+2\/— 5) = (6, 2 + 2^=5, 3 + 3 y— 5, i+y-^), since 2 + 2V^ r 5 — 6 = — 4 + 2V = T and 3 + 3 V^ — (2 + 2 %/— 5) = 1 + V^ whence, since all numbers in the symbol are multiples of 1 + V — 5, (2, 1 + V- 5) (3, 1 + V- 5) = (I + V- 5). Ex. 4. (2, i-f-y~5)( 3 , 1 — V— 5) = (6, 2 — 2 y— 1, 3 + 3^/^5, 6) = (6, 2 — 2^=5, 3+3-^—5; i—y—g), since 6— (2 — 2V= r 5) — (3 + 3V zr 5) = I — V 1 ^ whence, since 3 + 3V— 5 = 6 — (2 — 2 V— 5) — ( 1 — V— 1) . (2, 1 + y— 5) (3, 1 — V — 1) = (6, 2 — 2 y — 1, 1 — v— "5) = (i-y-5)> since all the numbers in the symbol are multiples of 1 — \J — 5. Ex. 5. Show that a) (3, 1 + 2V :=r 5) (3, 1 — 2V :zr 5) =5= (3), b) (7, 1 + 2V-TK7, I — 2\/— 5)-(7), c) (3, i + 2V-~5)(7, l+2\fi^$)=(l + 2y/^J), d) (3, 1 — 2\/ zr 5)(7, 1 — 2 V— 1) = (1 — 2 V— 5). THE REALM k(\/ 5). 263 § 12. Divisibility of Ideals. An ideal, a, is said to be divisible by an ideal, h, when there exists an ideal, c, such that a = Bc; b and c are then said to be divisors or factors of a. § 13. The Unit Ideal. Every ideal a, = (a x , a 2 , • • -,a m ), of &(V — 5) is divisible by the ideal (1), for ct( 1 ) = (a lt a 2 ,---,a n ) ( 1 ) = (a lf a 2 , • • •, a») = a. That (1) is the only ideal of fc(V — 5) possessing this property can be easily shown. Suppose that there is another ideal b= (8 X ,8 2 , •••,§„), which is a divisor of every ideal of fc(V — 5)- Since it divides the ideal ( 1 ) , we must have ( I ) = b'm, where m= Oi,f4, •••,**«•)• Then (1) = (S x , 8 2 , •••,8n)(/*i,/* 2 » •"•*f»»)j and hence 1 = g^ft, -f £A/% H 1- £ m „8„/>im 1 ) = AjSi -p- A 2 8 2 H~~ ' * * ~T~ A n 8 n , where i v i t , ■■■,$mn and hence a^a.,, --jAn are integers of &(V — 5). Therefore 1 is a number of b and b=(8 1 ,8 2 ,-..,8n,i) = (i). The ideal (1) is therefore the only ideal which divides every ideal of k ( V — 5 ) • Hence it is called the unit ideal of & ( V — 5 ) • It is evidently the whole system of integers of fc(V — 5)- It should be noticed that from 1 ) it follows also that 1 is a number of m, and in general we may show by this method that, if an ideal a be divisible by an ideal h then all numbers of a are num- bers of h. § 14. Prime Ideals. An ideal different from (1) and divisible only by itself and (1) is called a prime ideal. An ideal with divisors other than itself and (1) is called a composite ideal. 264 THE REALM &(V 5). We shall show that (2, 1 +V — 5) is a prime ideal. If this be not the case, two ideals, a and 5, neither of which is (1), must exist such that (2, i+V— 5) = aB. Let a = (a 19 a 2 , • • ., a m ), B = (ft, 0* • • •, 0„). Then we should have (2, i+V— l) = (a 15 a 2 , ...,a w )(^,^ 2 , ...,/? n ). It may be shown now by the method employed in the last paragraph that 2 and 1 +V — 5 are numbers of each of the ideals a and b, and hence (2, I + V— 5) = («1» ' • 'I Am, 2, I + V— 5) (ft, ...,j8„,2, i+V— 5). Let at, = a + & V — 5, be any one of the integers a lt a 2 , • • • , a m ; then ai = Z?(i+V — 5)+° — &• But a — b is a rational integer, and hence is of the form 2c or 2c + 1, where c is a rational integer. We have therefore either ai = fc(i+V— 5) + 2 ^ x ) or ai =^fc( I ^-yZT5)^2c-f 1. 2) If 1 ) be the case, ai may be omitted from the symbol a. If 2) be the case, we have ai — b{i+^~—^s)— 2c=i, and 1 may therefore be introduced into the symbol of a ; all other numbers could then be omitted and we should have a=(i). Proceeding in this manner with each of the numbers a lt a 2 , '•',a m , we see that one of the two following cases must occur, either all of the numbers a xi a 2 , '-,a m are linear combinations of 2 and 1 +V — 5, and hence may be omitted from the symool of a, in which case we have a=(2, i+V— ~5), or some number of a is not a linear combination of 2 and 1 + V — 5> m which case 1 may be introduced into the symbol of a and we have a=(i). THE REALM &(V 5)- 265 The same is evidently true for B. We have therefore as the only possible factorizations of (2, 1 +V — 5) (2, i+V=5) = (i)(i) = (i), 3) or — (2, 1 + V^) (2, 1 +V— 5), 4) or =(2, i+V=5)(i), or =(i)(2, i+V— ~5)- It has already been proved that (2, i+V-5) + (i), hence 3) is impossible. Likewise it may easily be shown that 4) is impossible, for we have seen (§11) that (2, i+y=5)==(2), while, since 1 +V — 5 1S not a multiple of 2, (2, i+v-l)4=(2). Hence 4) is impossible. The only divisors of (2, 1 +V — 5) are therefore the ideal itself and (1). Hence (2, 1 +y — 5) is a prime ideal. It may be shown similarly that (3, 1 +V — 5) and (3, r — V — 5) are prime ideals. The proof in these cases is sug- gested as an exercise. Ex. Prove that every ideal of the form (p, 1 -j- q\/ — 5), where p and q are rational primes different from each other, is a prime ideal. § 15. Restoration of the Unique Factorization Law in Terms of Ideal Factors. We shall now show that although the factorization of 6 into its prime number factors in k{ V — 5) * s not unique, nevertheless, when we resolve the principal ideal (6) into its prime ideal fac- tors this factorization is unique. 1 There are evidently two differ- ent factorizations of (6) into principal ideal factors; that is, (6) = (2)(3) = (i+V-5)(i-V^5). 1) 1 We speak of the factorization of an integer a into its ideal factors, meaning thereby always the factorization of the principal ideal (a) defined by a. 266 THE REALM k(\/ 5). These factors are, however, not prime ideals, for we have shown (§11) that (2) = {2, i+V^5) 2 , (3) = (3, i+V-l)(3, i-V-1), (i+V=5) = te r+V^5)(3, i+V-5), and (1— V— 5) = (2, i+V— 5)(3, i— V— 5)- We have shown also (§ 14) that these factors of (2), (3), (1 + V — 5) and (1 — V — 5) are a *l prime ideals. Substituting in 1) we have (6) = (2) (3) = (2, i+V-5) 2 (3. i+V=5)(3> 1— V^Di and (6)=(i+V-5)(i-V-5) =(2, i+V-5)(3, i+V— 5) (2, i-V— 5)(3, 1— V— 5) =(2, i+V— 5) 2 (3, i+V-5)(3, 1— V— 5). Hence (<5) can be factored in one and but one way into prime ideal factors. Ex. Show that the factorizations of 9, 14, 21, and 49 into prime number factors are not unique but that the factorizations of (9), (14), (21), and (49) into prime ideal factors are unique. We have now shown that the introduction of the conception of the ideal in &(V — 5) has accomplished, at least in the particular example given, what we desired; that is, the restoration of the unique factorization law. Instead of showing that the unique factorization law holds in general in &(V — 5) when the factorization is expressed in terms of prime ideal factors, and then investigating the properties of the integers and ideals of this realm, we shall proceed at once to the discussion of the general quadratic realm defined by the root of any irreducible quadratic equation. Among these realms are included, of course, the special realms k(i), &(V — 3), &(V 2 ) and &(V — 5). We shall see that when the factorization in any quadratic realm whatever is expressed in terms of prime ideal factors it is unique, and we shall be able to deduce general theorems for the integers and ideals of any realm similar to those THE REALM £(\/ 5). 267 found for the integers of realms in which the unique factorization law held in the ordinary sense. We shall find, moreover, that the introduction of the ideal will lead us to the discovery of new and deeper properties of these realms. The introduction of ideal factors is due to Kummer, but the form used in the text and known as ideals is due to Dedekind. For an account of Rummer's researches see his papers, Crelle, Vol. XXXV, pp. 319 and 327, especially the former, in which he announces his introduction of the ideal number; in the latter paper he expands the theory. A brief account of Rummer's con- ception is given in the eleventh supplement to Dedekind's edition of Dirichlet lectures, pp. 545-550; see also Bachmann, Allgemeine Arithmetik der Zahlenkorper, pp. 150-160, for a very interesting discussion of Kummer's ideal numbers and other methods of reinstating the unique factorization law in the general algebraic number realm. CHAPTER IX. General Theorems Concerning Algebraic Numbers. § i. Polynomials in a Single Variable. 1 Before beginning the study of the general quadratic realm we shall give a few theorems which are necessary for our future investigations. First of all, we shall prove a theorem concerning the divisibility of polynomials in a single variable. By a polynomial in a single variable, x, is meant, as has been said, an expression of the form a x n -f- a^- 1 + • • • + a n, where n is a positive rational integer and the a's are quantities not containing x. The sum, difference and product of two poly- nomials in x are evidently polynomials in x. In what follows we shall in all cases assume the a's to be rational numbers. A polynomial, f(x), is said to be divisible by another poly- nomial, /i(-r), when a third polynomial, f 2 {x), exists such that /<*)— /«.(*)£(*)• It is evident that all polynomials of the oth degree, that is, the rational numbers, divide every polynomial in x. If f x {x) and f 2 (x) have no common divisors other than con- stants, they are said to be prime to each other, or to have no common divisor. Theorem i. // f 1 (x) and f 2 (x) be two polynomials in x without a common divisor, there exist two polynomials in x, 4> x (x) and </> 2 (-r), such that 1 Weber : Algebra, Vol. I., §§ i to 6. 268 GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. 269 Let f ± (x) and / 2 (^) be of degrees m and n, respectively, and m ^ n. By division we may put f t in the form ft=*qjt+f» 1) where q 19 the quotient, and / 3 , the remainder, are polynomials in x, and / 3 is of lower degree than / 2 . Likewise we may put f 2 in the form /2 = 9 2 / 3 +/4, 2) where f z and f 4 are polynomials in x, and f 4 of lower degree than / 3 . Continuing this process, which is none other than that of finding the greatest common divisor of f x {x) and f 2 (x), we have f*=qJ*+U 3) U = ^^5 + /«> and arrive finally at a point where the remainder is a constant, / fc , different from o, since f x and / 2 are prime to each other. We have then fk- 2 = qk- 2 fk- 1 +fk. Putting now the value of / 3 from i) in 2) we have /4=(i+2i? 2 )/ 2 — qJi', that is L = r 1 f 1 + r 2 f 2 , where r, and r 2 are polynomials in x. Putting the expressions for / 3 and f 4 in terms of f x and f 2 in 3), we obtain T 5 === ^1/1 T ^2/ 2> where f^ s 2 are polynomials in ^r. Continuing this process, we obtain finally fk = W 1 f 1 + ZV 2 f 2 , where w lt w 2 are polynomials in x. As has been said, /& is a con- stant different from o. Putting therefore w t = A-4>i O) , W a = fk<l> 2 (x) , 270 GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. we have ftfc-(*)A"(*) +/i*i(*)/iOO =/*» and hence where ^(x) and <£ 2 (.*") are polynomials in x. We may generalize the above theorem as follows: Theorem 2. // A (■*') an ^ f* ( x ) ^ * wo polynomials in x without a common divisor and g{x) any polynomial in x, there exist two polynomials in x, ^(a*) and $ 2 (x), such that ® 2 ( x ) w of lower degree than f x (x) and •kWAW + •*(*)/.(*) — tft*)'i By Th. 2 there exist two polynomials in x, ^(x), </> 2 (a), such that ^)/iW+fc(')/iW = i 4) Multiplying 4) by ^(^) we have #0)4>iO)/iO) +#<*>**(•*)/*(*) =#<»• 5) Putting #(a-)<£ 2 (a") m the form ^W^W=?W/iW + '(*)» where #(a) and r(.r) are polynomials in x and r(.r) is of lower degree than f 1 (x), and substituting in 5), we have [g^)<f> 1 (x)+q(x)f 2 (x)]f 1 (x)+r(x)f 2 (x)=g(x); that is *d*)fd*)+*t*)fn(*)=9(*)* where $ x {x) and ® 2 {x) are polynomials in x, and ® 2 ( x ) * s °f lower degree than f x {x). A polynomial, /(a), is said to be irreducible in the realm k(a) when it cannot be resolved into integral factors whose coeffi- cients are numbers of k(a). When f(x) has rational coefficients and is said simply to be irreducible, no realm being specified, the rational realm is understood; that is, f(x) is not resolvable into integral factors having rational coefficients. Theorem 3. An irreducible polynomial, f(x), can have no factor in common with another polynomial, F(x), unless F(x) be divisible by f(x). GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. 27 1 The coefficients of the greatest common divisor of the two polynomials F(x) and f(x) are derived from the coefficients of these two polynomials by rational operations and are therefore rational numbers, since the coefficients of F(x) and f(x) are rational numbers. But f(x) is divisible by no polynomial in x with rational coeffi- cients except itself and the rational numbers. Hence either F(x) and f(x) have no common factor or F(x) is divisible by f(x). Cor. 1. If f(x) be irreducible and F(x) vanish for one root of the equation f{x) —o, it vanishes for all roots of f(x) =o. For, if F(x) vanish for a root of f(x) =o, F(x) and f(x) must have a common factor. But this can only be f(x). Cor. 2. If f(x) be irreducible and F(x) be a function of lower degree than f(x) that vanishes for one root of f(x) =o, then F(x) must vanish identically; that is, all coefficients of F(x) are o. § 2. Numbers of a Realm. Let us consider the realm k(a) of the nth degree, a being a root of the irreducible rational equation /O) = x- + a,*** H h *»=0, 1 ) whose remaining roots we denote by a', a", •••,a (n " 1) . Any number of k(a), being produced from a by repeated performance of the operations of addition, subtraction, multipli- cation and division, is a rational function of a with rational coeffi- cients and hence can be expressed in the form where x( a ) an d «A(a) are rational integral functions of a with rational coefficients. The realm k(a) is composed therefore of all rational functions of a with rational coefficients, the denomi- nator never being o. We shall now show that every number of the realm can be expressed as a rational integral function of a with rational coefficients. 272 GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. The degrees of x( a ) an d $(&) can be made lower than the nth by virtue of the relation a n + a^- 1 -\ f- a n = o. Since ^( a ) 1S different from o and of degree lower than the nth, \p(x) is not divisible by f(x), and hence, since f(x) is irreducible, if/(x) is prime to f(x) (Th. 3). We can therefore by Th. 1 find two polynomials in x, ^ 1 (x), & 2 (x), with rational coefficients and <£ 2 (.r) of lower degree than the wth, such that $ 1 W/M+$ 2 (^W= X W. 2) Putting a for x in 2) we have **(a)y(<x)— x(a), and hence that is, = & + & x a + fr 2 a 2 -] j- fc^a*" 1 , where b Q ,b x , ••• i b n _ x are rational numbers. This representation of is unique, for, if we had also = c + c % a + c 2 a 2 H l. Cn^a"- 1 , then it would follow that fro — ^0+ (fri — 0<H h (frn_x — c n _ 1 )a n - 1 = o; that is, a polynomial in ^r of degree lower than the nth would vanish for x = a, but this by Th. 3, Cor. 2 is impossible unless all the coefficients of the polynomial are o. Hence and the two representations are identical. The numbers of the realm are seen therefore to be coextensive with the totality of rational integral functions of a with rational coefficients and of degree not higher than the (w — i)th. We shall next prove the following simple theorem : GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. 2/3 Theorem 4. Every number 6 of k(a) satisfies a rational equation, whose degree is the same as that of the realm, and whose remaining roots are the conjugates of 0. Form the equation <!>(t) = (t — 0)(t — 0')'--(t — 6< n - 1 >) = t n + dit n-l+... + dn==0) 3) where 0', 0", • • •, #*** are the conjugates of 6. The coefficients, d lf d 2 , •••, d n , of 3) are symmetric functions of the roots of 1) and hence rational functions of the coefficients of 1). Hence d ± ,d 2 , -~>d n are rational numbers. Therefore $ satisfies a rational equation of the nth degree, whose remaining roots are the conjugates of 6. Every number of the realm is therefore evidently an algebraic number. We turn now to the reducibility of $(0> and shall prove the following theorem : Theorem 5. The function <£(£) is either irreducible or is a power of an irreducible function. The n conjugates of a number of k(a) are either all different or else fall into n x systems, each containing n 2 numbers all alike. In the first case, &(t) is irre- ducible, in the second, &(t) is th£ n x th power of an irreducible function of the n 2 th degree. If &(t) be reducible it must be a product of irreducible factors, each of which vanishes for one or more of the quantities 0,6', ••.,0< n - 1 >. Let *(0=*i(0*2(0---*»i(0> where <j> x {t) , $ 2 {t) , •••,<^>m(0 are irreducible and let <f> x {t) vanish for t = ; that is, <f> 1 (0)=o. We have seen that o=g(a), where a is the number defining the realm and g(a) a rational integral function of a with rational coefficients. Then <£i[#(s)]=o. 18 274 GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. The equations 4>i[#0')]=o and /(*)=0 have therefore a root in common, and, since f(x) is irreducible, <f>i[9( x )] must vanish for all roots of f{x) =o; that is, ^[^(^)]=o,^ 1 [^(a-)]=o,...,^ 1 [ 5 r(a^)]. But 0' = g(a'),e" = g(a"),.->,0^=g(a( n -v). Hence $,(6) =0,^(6') =o, -..^^"-v) =o; that is, £i(0 vanishes for all of the w conjugate numbers 0,0', •••,0 (n_1) . If these numbers be all different, <f> t (t) is of the nth degree and hence identical with $(0- If, however, there be among them only n 2 which are different from each other, say 6,0', ...,0<"«- 1 >, then <j> 1 (t) = (t — 6)(t — 6 f ) ••• (t — 0^-^). Since, moreover, every irreducible factor of ®(t) vanishes for one of the quantities 0, 6', •••, W_1 , and hence for all of them (Th. 3, Cor. 2), every one of these irreducible factors of &(t) is identical with <f> r (t) ; that is ^(Oj^sCO* •**»^n(0 are all iden- tical with ^(0- Therefore $(t) is in this case a power of <f> x (t) ; that is, $(t) = [<^>i(0] n S where %n 2 = n. We have seen (Chap. I, § i) that every algebraic number sat- isfies a single irreducible rational equation. We see now from the above that the degree of this equation is a divisor of the degree of the realm of which 6 is a number. According as the degree of this equation is the same as or lower than that of the realm, 6 is said to be a primitive or imprimitive number of the realm. Thus 6 is a primitive number of k(a) when ft is different from all of its conjugates and an imprimitive number when this is not the case. GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. 275 Theorem 6. Any primitive number of k(a) may be taken to define the realm; that is, k(0) = k(a). Let be any primitive number of k(a) and $', 0", •••,0 (n_1) its conjugates, and let w be any number of k(a) and <«/, <d", •••,o) (n ~ 1) its conjugates. We shall show that o> can be expressed as a rational function of with rational coefficients, and hence that k(0)=k(a). We have q>(t) = (t — 0)(t—0') ••• (*— ^ (n " 1) ). Then (ft) ft)' G)( n_1 ) \ 7zrg + — 3-, + ... + 7 -^ r) ) _*(,). 4 ) where ^(O is a polynomial in £ of the (« — i)th degree, whose coefficients are rational numbers, for they are symmetric func- tions of the roots of the irreducible rational equation satisfied by a, and hence rational functions of its coefficients. Putting for t in 4) we have <»(0 — f ){0 — 0") ■•.(0 — 0< n -v)=*(0), or, putting as usual d/dt>$>(t)=&(t) = (t — 0'){t — 0").-.(t — d<"- 1 >)+terms. containing the factor t — 0, we have where &(0) is a polynomial in t with rational coefficients, and is different from o, since is different from all its conjugates. Every number of k(a) can therefore be expressed as a rational function of with rational coefficients. Hence all numbers of k(a) are numbers of k(0), and therefore k(a)=k(0). Theorem 7. If f(x) =** + a x x n ~ x ^ f- a n = 5 ) be an irreducible rational equation, and 0, one of its roots, be an 276 GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. algebraic integer, the remaining roots, 0', 0", •••, (n_1) , are also algebraic integers. This theorem follows directly from Th. 4, Chap. II. It may also be proved as follows. Since is an integer, it must satisfy an equation F(x) = x n + b x x** H h ^ = o, 6) whose coefficients are rational integers. But if F(x) vanish for one root of the irreducible equation 5), it vanishes for all roots of 5). Hence 6', 6", • ••,0 (n - 1) satisfy 6) and are integers. Theorem 8. The sum, difference, product and quotient, the denominator of the latter not being zero, of two algebraic num- bers are algebraic numbers. Let a and /? be two algebraic numbers, which satisfy respect- ively the two irreducible rational equations x m + a^™- 1 -\ h a m = 0, 7) ,*• + fr^H \-b n = o. 8) The necessary and sufficient condition that a + (3 shall be an algebraic number is that it shall satisfy a rational equation. Form the equation [ x —(a + p)] ••• [(*— .(a«>+0<»)] ••• [*— (a ( «- 4) +j8<«- 1 >>] =X mn + Cl X mn ^ H \- Cmn = 0, 9) whose roots are the mn numbers f a = a,a', •••,a (w - 1) , a + £' J /3 = p,(3',---,/3< n - 1 \ ■ The coefficients c lt c 2 , •••,c mn of 9) are symmetric functions of the roots of 7) and 8), and hence rational functions of the coeffi- cients of 7) and 8). But the coefficients of 7) and 8) are rational numbers. Hence the coefficients of 9) are rational numbers, and a-\-/3 is therefore an algebraic number. The proofs for a — /?, a/3 and a/f3 are of the same character. Cor. 1. Every rational function of any number of algebraic numbers with rational coefficients is an algebraic number. GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. 277 Cor. 2. The sum, difference and product of two algebraic in- tegers are algebraic integers; for in this case the c's being not only rational but integral functions of the a's and b's, and the a's and b's being now integers, the c's are themselves rational integers. Cor. 3. Every rational integral function of any number of algebraic integers with rational integral coefficients is an algebraic integer. We obtain a still more general theorem when we notice that, if we allow the coefficients b x , b 2 , --,b n of the equation X* + ^«-i _| (_ 0n = IO ) to be any algebraic numbers instead of restricting them to rational numbers, the roots of 10) will nevertheless be algebraic numbers. Theorem 9. If & be a root of the equation F(x)=.r n + a 1 x n -i-{ \-a n = o, where a x ,a 2 ,--,a n are any algebraic numbers, it is itself an algebraic number. Let a x , a 2 , '-,a n satisfy rational equations of degree m x , m 2 , • • •, m n , respectively, and let the remaining roots of these equations be a ' a " ... rt t*"*- 1 ) Let m — m x m 2 ••• m n and form by putting for a< a*, a/, •••, a (mi-i) (i— i } 2, ••-, n) the m polynomials in x F(x) =x n + a x x n -* H h a n , F x (x) =x n + a x 'x n -i H \-a n , F 2 (x) =x n + a/ V- 1 H h a», F m _ x (x) =x n + a/^- 1 ^"- 1 H h a„ (m »- 1) . Form the product FF x F 2 .-.F m _ x = f(x). 278 GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. The coefficients of f(x) will be symmetric functions of the roots of the rational equations satisfied by a 19 a 2 , • • • , a n , and hence rational functions of their coefficients. They are therefore rational numbers and <o, being a root of the rational equation is an algebraic number. Ex. 1. Let w be a root of the equation F(x) = x 2 + V 2* + y/J= o. 11) We see that V 2 an d V3 are roots respectively of the rational equations x 2 — 2 = and x' 1 — 3 = o, whose remaining roots are — V 2 an d — V3- We have FxO) = x 2 + \/2X — V3T F 2 O) = x 2 — \fex + V3> F 3 (x) —x 2 — yj2x — V3~i and f(x) =F F!F 2 F z = x s — 4x* — 2x i — 12^ + 9 = 12) Hence, w being a root of 12), is an algebraic number. It is moreover an integer, since the coefficients of 11) are integers (see Cor. 1 below). Cor. I. If to be a root of the equation F(x)=x n + a 1 x n ~ 1 + ••• -j-a n = o, where a lt a 2 , '-,a n are algebraic integers, it is itself an algebraic integer; for the coefficients of f(x) formed as above are not only rational but integral functions of the coefficients of the rational equations satisfied by the a's and these are now rational integers. Hence the coefficients of f(x) are rational integers, and o> is an integer. Theorem 10. Every algebraic number can by multiplication by a suitable rational integer be made an algebraic integer. Let the algebraic number, a, be a root of the rational equation and let a be the least common denominator of the a's. Then a n + -J • a 71 ' 1 + - 2 • a 11 - 2 + . . . + -= = o, 13) where the b's are rational integers. GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS. 279 Multiplying 13) by a n , we have (a.aY + ^(0,0.)^ + a Q b 2 (a a) n -* +'•• +V- 1 &» = o; that is, a a is a root of the equation y n + bj*- 1 + a b 2 y n -* H f- ao"-^, = o, whose coefficients are rational integers, and is therefore an alge- braic integer. Ex. Let a be a root of •* 3 -hf* 2 + f* + ! = o, that is, of * + T V 2 .+ H* + tI = o. J 4) Multiplying 14) by 12 3 , we have (i2x) 3 -\- 6(i2x) 2 -\- 192(12*) +2160 = 0. Thus 12a is a root of the equation y 3 + 6y 2 + 192V + 2160 = o, and hence an integer. This is seen to be simply the transformation of 13) into an equation whose roots are a times those of 1), a being selected so as to make the coefficients of the new equation integers. CHAPTER X. The General Quadratic Realm. § i. Number Defining the Realm. By the general quadratic realm we understand the realm de- fined by a root of the general irreducible quadratic equation of the form ax 2 + bx + c = o, I ) where a, b and c are rational integers. If a be a root of i), this realm is denoted by k(a). If a' be the other root of i), the realm k(a') is the conjugate realm of fc(a)(Chap. I, §4). Solving i), we have — b + V& — 4ac , — b — V b* — ^ac a = , a = 2a 2a Put b 2 — 4ac = l 2 m, where m contains no square factor ; then V& 2 - — 4ac = iym, and k{a) =k(ym) for a — b -}- / V m 2a is evidently a number of fc(Vm) and — 2aa -f b \/ *n : / is a number of k(a). Hence k{a)=k{ym). 1 Hence, to consider all quadratic realms, it is sufficient to con- sider all realms defined by a root of an equation of the form x 2 — m = o, 2) 1 See Chap. IX, Th. 6. 280 THE GENERAL QUADRATIC REALM. 28 1 where m is any rational integer containing no squared factor. We shall understand in what follows by ~\Jm the positive real or imaginary root of 2), and shall assume that m contains no square factor. The conjugate realms k(a) and k(a') are identical, since a is evidently a number of k(a') and a' a number of k(a). The general quadratic realm is the simplest example of what is known as a Galois realm; that is, one which is identical with all its conjugate realms. §2. Numbers of the Realm. Conjugate and Norm of a Number. Primitive and Imprimitive Numbers. Let a be a root of the irreducible quadratic equation * 2 + P x + Q. — °- Every number, w, of k(a) is a rational function of a with rational coefficients, and hence has the form a -f ba (0=z — , c + a a where a, b, c and d are rational numbers. a -f ba The number co' = c + do! obtained from <o by the substitution of a! for a is the conjugate of w (Chap. I, § 4). The numbers of k(a) that are rational are seen to be their own conjugates. We shall show now that every number, w, of k{a) can be put in the form co = e + fa* where e and / are rational numbers. 1 First, let abe -\/m. Then we have c + dV 1) m "See Chapter VIII, §2, for general theorem of which this is a special case. Simplified proofs are given here of this and several following theorems. 282 THE GENERAL QUADRATIC REALM. Multiplying the numerator and denominator of i) by c — d^Jm, we obtain ac — bdrn be — ad ,— ~~ c 2 — d 2 m c 2 — d 2 m All numbers of k(\/m) can therefore be put in the form e + /V ra, where e and / are rational numbers. If w, = a-[-frVw > be any number of k(-\/m) it satisfies the quadratic equation x 2 — 2ax-\-a 2 — mb 2 = o, 2) whose other root is uy', = a — b^Jm, the conjugate of w. Hence every number w of k(^/m) satisfies a rational equation of the second degree (Chap. IX, Th. 4). We say that a is a primitive or imprimitive number of k(^/m) according as the equation 2) is irreducible or reducible. The necessary and sufficient condition for 2) to be irreducible is evidently b=^=o. In other words, a is a primitive number if it be different from its conjugate (Chap. IX, Th. 5). If b = o, and hence w = w ' = a, then w satisfies the rational equation of the first degree x — = 0. The primitive numbers of a realm are thus seen to be those defined by equations of the same degree as that of the realm (Chap. IX, Th. 5). The imprimitive numbers of a quadratic realm are evidently the rational numbers. If « be a primitive number of a realm of the wth degree and the identity Go + ai« -j (- an-iw"- 1 = b + bi<a-\ + frn-iW*" 1 3) exist where the a's and b's are rational numbers, then the coefficients of the same powers of « in the two members of 3) must be equal; that is, a =bo, di = bi, •••, a»-i = &n-i ; for otherwise « would satisfy an equation of degree lower than the wth, which is contrary to the assumption that w is a primitive number of the realm. We have shown (Chap. IX, Th. 6) that any algebraic number realm can be defined by any one of its primitive numbers. This THE GENERAL QUADRATIC REALM. 283 can be proved for the special case of quadratic realms very simply as follows: Let a be a primitive number and w any number of &(\/m). We have seen above that a and w can be put in the forms a = a-\-b-\/m, 4) (o = c + dy/m, 5) where a, b, c and d are rational numbers. From 4) we have ,— a — a 1/111 — Vm ~ b ' and from 5) be — ad d — b +- d a. Hence every number <o of k(^/m) can be written in the form w = e + fa, where e and / are rational numbers and a a. primitive number of fc(V w )- Hence and we have proved not only that every quadratic realm may be defined by any one of its primitive numbers, a, but that every number, w, of the realm k(a) may be put in the form < 1 > = e + fa, where e and / are rational numbers (Chap. IX, § 2). We may evidently choose as the primitive number defining the realm an integer. In what follows we shall suppose this to have been done. The product of a number, w, of k(a) by its con- jugate a/ is its norm 1 and is denoted by «[<u] ; that is, w[to] =toto'. Since n[w] is a symmetric function of the roots of the rational equation satisfied by a, it is a rational function of the coefficients of this equation, and hence a rational number. In particular when the realm is defined by V w, we have w[to] = (a-\-b-\/m)(a — b}/m) =a 2 — b 2 m. 'Hilbert: Bericht, §3. 284 THE GENERAL QUADRATIC REALM. §3. Discriminant of a Number. 1 The square of the difference of a number a and its conjugate is called the discriminant of the number and is denoted by d[a] ; that is, ^[a] ==(<*- a') 2 = It is evidently a rational number and the discriminant of the quadratic equation x 2 + a x x + a 2 = o, whose roots are a and a'. If a be a primitive number of the realm its discriminant is different from o, and conversely, if d[a] be different from o, a is a primitive number. § 4. Basis of a Quadratic Realm. Theorem i. There exist in every quadratic realm two in- tegers, »!, w 2 , such that every integer, <o, of the realm can be expressed in the form (a = a 1 o) 1 -j- a 2 a) 2 , where a x a 2 are rational integers. 2 Suppose the realm to be defined by an integer, a, a supposition in no way limiting the generality of the proof, and let w be any integer of k(a). By the preceding paragraph w can be put in the form <» = r 1 + r 2 a, 1) where r t and r 2 are rational numbers. We have o> —r x -\-r 2 a!. 2) Solving 1) and 2) for r x and r 2 by means of determinants, we have 'Hilbert: Bericht, §3. 2 Hilbert : Bericht, Satz 5. This proof could have been somewhat sim- plified had greater use been made of the fact that the realm under con- sideration was quadratic, but it seemed desirable to give the proof in a form at once extendable to realms of any degree. THE GENERAL QUADRATIC REALM. 285 r, = ft) a 1 w a |I a\ ft)' a! |v a' |I «'l I a I 1 -I 2 I a! 1 1 a' I ft) 1 I a I ft)' l« a' 1 I a 2 |1 a' *M' d\ay where A x and ^4 2 are rational integral functions of the integers <o, a, a and a' with integral coefficients and hence integers (Chap. IX, Th. 8, Cor. 3). Moreover, d [a] is a rational number and hence A lf = r 1 d[a], and A 2i = r 2 d[a], are rational numbers. Therefore, A x and A 2 are rational integers. Hence every integer, o>, of k[a] can be put in the form A + A 2 a where ^ t and ^4 2 are rational integers and J[a] is the discrimi- nant of a. Suppose, now, all integers of the realm to be written in the form 3) and consider those in which A 2 is not equal to o. Among these there will be some in which A 2 will be smaller in absolute value than in any of the remaining ones. A x ' + A 2 'a 3) Let ft>„ = rf[a] be one of these. Then A 2 will be the greatest common divisor of the values of A 2 in all integers of the realm; for if this be not the case, let A x " + A 2 "a be any integer such that A" is not divisible by A 2 , and let A be the greatest common divisor of A 2 and A 2 " . Then we can find two rational integers a and b such that aA 2 ' + bA 2 " = A, I . 286 THE GENERAL QUADRATIC REALM. and hence 7 = ™ 2 + ^ 3 = -^ aA/ + 6A t " + Aa is an integer in which the coefficient of a is less in absolute value than A 2 , which is contrary to the supposition that there is no value of A 2 less in absolute value than A 2 '. Hence A 2 = a 2 A 2 , where a 2 is a rational integer. Denoting « — a 2 w 2 by to*, we have A x + A 2 a - <y4/ - a 2 A 2 f a A x - a 2 A/ d\a\ d[a] ' Consider now those integers of the realm in which A 2 = o, butA^o. 1 There will be one or more among them in which A x is less in absolute value than in any of the remaining ones. Let <o 1 = A 1 ,,, /d[a] be one of them. We see as above that A™ is the greatest com- mon divisor of the values of A t in all the integers in which A 2 = o, ^/4=o, 2 and hence <o* = to — a 2 o) 2 = d^x, or 0) = Ojtoi -j- t7 2 co 2 . 4) There exist, therefore, in every quadratic realm two integers, <a lf eo 2 , such that every integer, w, of the realm can be expressed in the form 4), when a lt a 2 are rational integers. ^he remainder could be worded much more simply, if the fact that (A t — 0^4/) /d [a] is a rational integer be made use of, but the above form seems better as it is in line with the general theorem. 2 The integers, in which A*=zo and Ai=^=o, are evidently the rational integers, excluded. Also A"' = d[a], and w x — 1. We have A x — a 2 Ai = a^i", where a t is a rational integer. THE GENERAL QUADRATIC REALM. 287 Every pair of integers, w ls » 3J possessing this property is called a basis of k(a). Cor 1. // » XI w 2 be a basis of k(a), then w/, « 2 ' is a basis of the conjugate realm k(a'). Theorem 2. If m lf w 2 be a basis of k(-\/m), the necessary and sufficient condition that • * 1 W 2 * = b^-L + ^2 W 2> w/^r*? a lf a 2 , & 1? & 2 are rational integers, shall be also a basis of k(^/m) is K K — db I, For the proof of this theorem see the corresponding one in k(i) (Chap. V, Th. 1). § 5. Discriminant of the Realm. If m 19 w 2 be a basis of &(\/m), the square of the determinant formed by these integers and their conjugates is called the dis- criminant of the realm and is denoted by d; that is, 13 *-P "' 2 We see that d is a rational integer, for it is an integral sym- metric function of the roots, V w, — y/m, of the equation x 2 — m = o, and hence 'a rational integral function of the coefficients of this equation, which are rational integers. That the value of d is independent of the basis chosen may be shown as in k(i). The discriminant of every integer of the realm is divisible by the discriminant of the realm; for, if a = a 1 (o 1 -j- a 2 o) 2 , "Hilbert: Bericht, p. 181. 2 Hilbert: Bericht, p. 194. 288 THE GENERAL QUADRATIC REALM. be any integer of k(^/m), and i = b 1 o> 1 -J- b 2 a> 2 , then d\a\ = K w\ a l a t\ = ^V. d[a]=d, K K = d= a l a 2 CO, If then and i, a, is a basis of the realm. We see, moreover, that when d [a] is not divisible by the square of a rational integer, we have Q d[a]=d, and hence I, a, is a basis. 1 The converse of this theorem is, however, not true; that is d[a] may be divisible by the square of a rational integer and still i, a, be a basis. 1 The definition and deductions of this paragraph are immediately ex- tendable to the general algebraic realm of the wth degree. The last fact mentioned is of especial importance as it may be shown by the method used in the text that, if be a root of X n + diX n ~ x -\ + a n = o, where a u •••, a n are rational integers, and d[0] be not divisible by the square of a rational integer, then I, #, ..., n_1 is a basis of k(B). The great value of this fact is that although we may by the method of § 4 prove the existence of a basis in a realm of the nth degree, we have, however, general methods of determining a basis only in the cases of » = 2 or 3. The case n = 2 will be discussed in the next paragraph ; that for n = 3 will be found in Woronoj : The Algebraic Integers which are Functions of a Root of an Equation of the Third Degree, this being a translation of the Russian title. A short account of this method will be found in : Taf el der Klassenanzahlen fur Kubische Zahlkorper, by the author. THE GENERAL QUADRATIC REALM. 289 Thus in k(i)> d[i], = — 4, is divisible by 2 2 , but I, its a basis oik(i). § 6. Determination of a Basis of k(\/m). We have seen that every number of k(^/m) can be written in the form aWi + fyv/w, where r t and r 2 are rational numbers. Let r x = a/c, and r 2 = b/c, where c is the least common multiple of the denominators of r x and r 2 , r x and r 2 being in their lowest terms. Then a = ■, 1) c ' where a, b and c are rational integers having no common factor. The necessary and sufficient condition that a shall be an integer of k(^/m) is that it satisfy an equation of the form x 2 -f- px + q = o, 2) where p and q are rational integers, the other root of 2) being the conjugate of a; that is, a' = a — bym Hence we have as the necessary and sufficient conditions that a shall be an integer of k(^m) a + a' = — a a rational integer, 3) # 2 — mb 2 « . x aa r = 2 — = a rational integer. 4) Remembering that a, b and c have no common factor, and m no square factor, we shall show that c can have a value different from 1 only when m = 1, mod 4, and then can take only the value 1 or 2. '9 29O THE GENERAL QUADRATIC REALM. i. Let c = pc 1} p being a prime different from 2. Then from 3) it follows that a = o, mod p, and from 4) that a 2 — mb 2 = o, mod p 2 , and hence mb 2 = o, mod p 2 . 5) But 5) is impossible since m can not contain the squared factor p 2 , and if b were divisible by p then a, b and c would have a com- mon factor p. Hence c can contain no prime factor different from 2. s ii. Let c = 2 e . We can prove exactly as in i that e can not be greater than 1. Let e— 1 ; that is, c = 2. Then from 4) it follows that a 2 — mb 2 = o, mod 4, 6) From 6) we see that a can not be even, for this would require a 2 = o, mod 4, and hence m& 2 = o, mod 4, from which it would follow that either m contains the squared factor 2 2 , or a, b and c have the common factor 2. Hence a = 2a x + *• Likewise b = 2b x + 1 ; for b even gives & 2 = o, mod 4, and hence from 4) a 2 ==o, mod 4, which we have seen to be impossible. We see therefore that, if c = 2, a and b must both be odd in order that a may be an integer; that is, a = 2a 1 -\-i and b = 2b ± -\- 1. We must now determine the form that m must have in order that a 2 — mb 2 may be divisible by 4; that is, that c may be 2. From a = 2a x -f- 1 and b = 2b \ + I it follows that a 2 = 1, mod 4, and & 2 == 1, mod 4, THE GENERAL QUADRATIC REALM. 29 I and hence from a 2 — mb 2 z==o, mod 4, it follows that 1 — m = o, mod 4. 7) Therefore a and b odd and w= 1, mod 4, are the necessary and sufficient conditions that a 2 — mb 2 may be divisible by 4. We can have therefore c = 2 when and only when these conditions are satisfied. Hence, when m=i, mod 4, every integer, a, of &(Vra) has the form a -f- b\/m a — * where a and & are both odd or both even, and every number of this form is an integer of &(\/w). When w = 2or3, mod 4, the condition 7) not being satisfied, c can not equal 2, and every integer of k(^/m) has the form a = a-{- bym, where a and b are rational integers. Every number of this form is evidently an integer of k{^/m). The cases m=l, 2 or 3, mod 4, include all possible forms of m, ra = o, mod 4, being excluded, since m would then contain a squared factor. These three cases are illustrated respectively by the realms &(V — 3), &(V2) and k(y^i). We shall now show that, if w represent s/m, V ra or ( 1 + ■\/m)/2 f according as m = 3, 2 or 1, mod 4, then all integers of k(^/m) can be expressed in the form a = u + V(o, where u and v are rational integers. This is at once evident when w = 3or2, mod 4. To show it when w= 1, mod 4, we observe first that 1 -f Vm co = — 2 is then an integer, for it is of the form {a-\-byJm)/2, where a and b are both odd. a + bVnt Then, if a. = 292 THE GENERAL QUADRATIC REALM. be any integer of k(^m) (ra=i, mod 4), we have, since ■\/m = 2(t> — 1, a -f b(2(o— 1) a — b a = ^ = f- baa ; 2 2 that is a = u + Vm, where u=(a — b)/2, v=b are rational integers; for a and b are rational integers, and (a — b)/2 is an integer, since a and b are both odd or both even. Examples. 1. Give a basis of each of the following realms : &(V5),&(\/6), k(\/— ¥), fc(V— "13), &(V75) and £( V^i). 2. Tell whether each of the following pairs of numbers is a basis of the realm to which it belongs, 2 -|- 3 \/6, 1 -j- -y/6; 1 -+- V°\ 7 + 6V6; K3 + 7V5), i(-i — 3VS). CHAPTER XI. The Ideals of a Quadratic Realm. § i. Definition. Numbers of an Ideal. An ideal of a number realm is a system of integers, a lt a oy a 3 , •••, of the realm infinite in number and such that every linear combination, A^ + A 2 a 2 + A 3 a 3 -| ,of them, where A x , A 2 , A 3 , • • • are any integers of the realm, is an integer of the system. 1 The integers of the infinite system which constitutes the ideal are called the numbers of the ideal. § 2. Basis of an Ideal. Canonical Basis. Principal and Non- Principal Ideals. Theorem i. There exist in every ideal a of a quadratic realm two numbers, i x , i 2 , such that every number of the ideal can be expressed in the form i = l lLl -|_ / 2 t 2 , where l x and l 2 are rational integers. Suppose all numbers of a to be written in the form t = a 1 o) 1 -f- a 2 o) 2 , where <o lf w 2 is a basis of the realm, and consider those for which a 2 =f=o. Among them must be some in which a 2 is smaller in absolute value than in any of the remaining ones. Let t 2 , = b(* x -[- c <»2> De one of these ; then c will be the greatest common divisor of the values of a 2 in all the numbers of a (see Chap. X, Th. i). We have a 2 = l 2 c, x The definition given in &(V^~5) w iU be seen later to coincide with this. See also Hilbert : Bericht, p. 182. 293 294 THE IDEALS OF A QUADRATIC REALM. where l 2 is a rational integer, and hence i — 1 2 l 2 = (a x — l 2 b ) m%' Consider now those numbers of a in which a 2 — o, but a t =f=o. Just as before we can show that there exists among them cer- tainly one, i 1 = ao) 1 , such that a is the greatest common divisor of the values of a x in all the numbers of the ideal for which a 2 = o, a x =4= o. Hence a x — l 2 b = l x a, where l x is a rational integer, and we have i — l 2 h =: h l i} that is i = l x i x -\- 1 2 l 2 , hence i x , i 2 are the desired numbers. Any pair of numbers of a such as ij, i 2 , having the property required by the theorem, is called a basis of the ideal a. The nec- essary and sufficient condition that any other pair of numbers of a h* = a ih + a 2^ t 2 * = b x i x + b 2 t 2 , shall be a basis of a is that K K dz I This condition can be satisfied by an infinite number of sets of rational integers, a lf a 2 , b lt b 2 , and hence each ideal has an infinite number of bases. We shall call the particular basis a<a Xi b<a x + co> 2 defined as above a canonical basis. Taking i,« as a basis of the realm, we have as a basis of c^a, b + c<o, an especially convenient form, in which a is evidently the rational integer smallest in abso- lute value occurring in a. Cor. i. // a x (o x -J- a 2 w 2 , b x o) x -\- b 2 <a 2 and c x <a x -\- c 2 o) 2 , d x m x + d 2 w 2 be bases of the same ideal, then b i b 2\ d i d 2 1 See Chap. V, Th. i. THE IDEALS OF A QUADRATIC REALM. 295 Cor. 2. // a 1 o) 1 + a 2 w 2 , b 1 o) 1 -f- b 2 <o 2 be a basis of an ideal, a, and c 1 o) 1 -f- c 2 <a 2 , d 1 w 1 -j- d 2 <o 2 be any two numbers of a, and d x d 2 K K then c x <a x -\- c 2 to 2 , d 1 (a 1 -j- d 2 <a 2 is also a basis of a. Th. 1 shows at once that all ideals of a quadratic realm would be obtained, if we paired the integers of the realm in all possible ways and took each pair a, j3, as defining an ideal (a, /3) ; for among these pairs would be certainly a basis of every ideal of the realm. In this pairing, however, each ideal would be repeated an infinite number of times. The definition given of an ideal (§ 1) holds for realms of any degree, as does a theorem similar to Th. 1 : namely, in every ideal of a realm of the nth degree there exist n integers, H, h, • ••,% such that every number of the ideal can be expressed in the form hii -f- &*« + • • • + l^n, where h, U, •••, In are rational integers. See Hilbert: Bericht, Satz 6. If a lt a 2 , • • •, a r be r numbers of a such that every number of a can be represented in the form \ x a 2 + A 2 a 2 + • • • + XrCLr, I ) where a^ A 2 , ---yXr are integers of the realm, we can define a by the symbol (a 15 a 2 , --^ar) ; that is, we write a= (a ly a 2 , --,a r ), understanding thereby the infinite system of integers of the form 1), the A's taking all possible values. We shall call a lf a 2f --^CLr the numbers defining the ideal a. The numbers of a are all those of the form I ) . We may intro- duce into the symbol any integer which is a linear combination of those already there without changing the ideal defined by it. Thus, if a s = \ t a x + A 2 a 2 -| 1- \ r CL r , we have a= (a lf a 2f •••,a r ) = (a ± ,a 2 , ••-,a r ,a,) ; for the system of integers K a i + K a 2 H + ^rCLr is coextensive with the system A^i + A 2 a 2 + ' * ' + ^cir + A s a 8 , the A's taking all possible values. 296 THE IDEALS OF A QUADRATIC REALM. Likewise, if any integer in the symbol be a combination of the remaining ones therein, it may be omitted from the symbol. Thus, if a x = X 2 a 2 + A 3 a 3 + • • • + A r a r , we can write a = (a lt a 2 , • • •, a r ) = (a 2 , • • •, a r ) . We speak for the sake of brevity of (a lt a 2 , - • • , a. r ) as the ideal a, and instead of saying that we introduce a number, a s , into the symbol of a or omit it from the symbol, say that we introduce a 8 into the ideal a or omit it from the ideal, although a 8 is and remains a number of a. It will be convenient also, if i lf t 2 be a basis of a, to speak of (i 1? i 2 ) as a basis representation of a. The determination of the question whether an integer a belongs to a given ideal a will be greatly simplified by some properties of ideals which will be developed later. It can, however, be easily decided now, if we have a basis of the given ideal, for if &> = a i + a 2 w \ De an y integer of the realm and b x + b 2 u, c i + C 2 W be a basis of a, the necessary and sufficient condition that a shall be a number of a is evidently that two rational integers l x , l 2 exist, which satisfy the equation hiPx + ^2 W ) + h( c i ~\~ c 2<°) =^1 + 2 w - 2 ) Equating the coefficients of the powers of o> in the two mem- bers of 2), we obtain the equations b 1 l 1 + c 1 l 2 = a 1 , b 2 l 1 -\-c 2 l 2 = a 2 , 3 ^ which determine l x , l 2 . If the values of l lt l 2 found from 3) be integral, a is a number of a, otherwise not. If we have not found a basis of a, we can generally determine whether a is a number of a by means of the fundamental condition that a is or is not a number of a according as a is or is not a linear combination of the numbers defining a with coefficients which are integers of the realm. For an ex- ample of this method see p. 259. 1 Unless the contrary be stated, 1, w is taken as a basis of the realm. THE IDEALS OF A QUADRATIC REALM. 297 An ideal which consists of all and only those numbers of the form \a, where a is a particular integer and A any integer of the realm, is .called a principal ideal and is denoted by (a). An ideal not having this property is called a non-principal ideal. For examples of principal and non-principal ideals see Chap.VIII, § 10. It should be observed that although all numbers of a prin- cipal ideal, (a), are multiples of the single integer a, when as multiplier we take any integer of the realm, nevertheless, just as in the case of a non-principal ideal, a basis of (a) consists of two integers, aoi t , ato 2 , where a^, <o 2 is a basis of the realm, for every number of (a) has the form {a 1 (o 1 -j- a 2 o> 2 )a = a x OLu> 1 -f- a 2 CLo) 2) where a lf a 2 are rational integers. For example: a basis of (i -fV — 5) is * + V — 5> (i +V — 5~}x V="5 ; that is, i + V^5, — 5 + V-^T- If the difference of two integers a and /3 be a number of the ideal a, this fact is expressed symbolically by writing a = ft, mod a, 4) and we say that a is congruent to f3 with respect to the modulus a. The fact that a — (3 is not a number of a is expressed symbol- ically by writing a 4=/?, mod a, 5) and a is said to be incongruent to /? with respect to the modulus a. Every number, a, of the ideal a is congruent to o with respect to the modulus a, or in symbols a = p, mod a. 6) No meaning other than the symbolic expression of the facts mentioned must be attached for the present to 4), 5) and 6). Thus we write 3 — 2V— I^i + sV^, mod (7, S+V^), since 3 — 2\/^5— i 1 + 2 V— ~5) =2 — 4V— 5 is a number of (7, 3 +V — 5), and we write i + 5V^5 + 2 — 3V— 5, mod(i+2V =r 5), 298 THE IDEALS OF A QUADRATIC REALM. since 1 + 5 V— 5 — (2 — 3V~ 5) =— * + 8V~ 5 is not a number of (1 + 2 V — 5)- Although the actual determination of a basis of any given ideal of a quadratic realm must be postponed until the properties of ideals have been more fully investigated, we can, however, now determine whether any two given numbers of an ideal a are a basis of a. The necessary and sufficient condition for a lt a 2 to be a basis of the ideal a,= (a lf a 2 , •••, a r ), is evidently, since every num- ber of a has the form A^ + A 2 a 2 + • • • + A r a r , that for every possible choice of the A's we shall be able to find two rational integers, I lt l 2 , such that Ai^i + A 2 a 2 -\ f- XrOLr = tfo + l 2 CL 2 . j) Let w 1} a), be a basis of the realm, and &i = ciiOi 1 -\- biO) 2 "1 \i = CiWi -f- C?iW 2 J We have on equating the coefficients of the number defining the realm in the two members of 7) two equations between rational integers, whose satisfaction by suitably chosen rational integral values of l XJ l 2 for all possible choices of the c's and d's is the necessary and sufficient condition that a lt a 2 shall be a basis of Q. * Ex. 1. That 3, 1 + V — 5 is a basis of (3, 1 + V — 5) may be easily- shown by the above method. Every number of (3, 1 + V — 5) has the form (c 1 + rf 1 V^5)3+(c 2 + ^V^ r 5)(i + V^5), 8) where Ci, di, c«, d 2 are rational integers. If 3, i + V — 5 be a basis of (3, i + V — 5), then every number of the form 8) must be expressible in the form hs + / 2 (i + V — 5)> where h, U, are rational integers, and hence for every possible choice of Ci, d u c 2 , d 2 , we must be able to find rational integral values of h, U, which satisfy the equation (ft + <W^5)3 + ic, + hyp-*) (1 + V— S) = h3 + 4(1 + V=l), or 3ft + cz — $d 2 -f- (3di + C2 + d 2 ) V^ = 34 + h + hy/^J. 9) THE IDEALS OF A QUADRATIC REALM. 299 Equating the coefficients of the different powers of V — 5, we have $Ci + c 2 — 5^2 = 3/1 + U 10) 3di + c 2 + d 2 = U, 11) as the two equations whose satisfaction by rational integral values of It, h for every possible choice of Ci, di, c 2 , d 2 is the necessary and sufficient condition that 3, 1 + V — 5 shall be a basis of (3, 1 + V — S)- Sub- tracting 11) from 10), we obtain 3*i — 3di — 6d 2 = 3/1, 12) Zdi -\-c 2 -\-d 2 = U, a system equivalent to 10), 11), and which evidently fulfils the required conditions. Hence 3, 1 + V — 5 is a basis of (3, 1 + V — 5)- In this particular case, we might have arrived at the result by simply observing that 1 + V — 5 must be the required basis number b + cyj — 5, since c has in 1 + V — 5 the smallest possible value; that is, 1. Moreover 3 must be the basis number a, for if (3, 1 + V — 5) contain a rational integer smaller in absolute value than 3, it would contain 1 and we should have (3, i + V"= r 5) = d), that is easily shown to be impossible, the equation (ft + ca/^5)3 + (di + <W^~5) (1 + V^) = 1 not being satisfied by rational integral values of Ci, c 2 , di, d 2 . Therefore 3, i + V — 5 is a canonical basis of (3, 1 + V — 5). Having shown that 3, 1 + V — 5 is a basis of (3, 1 + V — 5)> we know that the necessary and sufficient condition for any two numbers, S t 2 , to be a basis of (3, 1 -f- V — 5) is that where <h$ a 2 , bi, b 2 are rational integers satisfying the condition |*i **| ==±i. Pi K 1 This condition is evidently satisfied by an infinite number of sets of values of (h, a 2 , b u b 2 , from which we obtain by 13) an infinite number of different bases of (3, 1 -f- V — 5)- Thus since II 4 we see that 3.3 + 1 . (1 + V — 5) = io + V — 5 11.3 + 4(1 + V— 5) =37 + 4 V^S is a basis of (3, 1 + V — 5). 3oo THE IDEALS OF A QUADRATIC REALM. On the other hand ii— 4V^ is not a basis of (3, 1 + V 2 5 = 2.3 — i(i + V — 5), 5=5-3— 4(l + \F"S). -5), since - I I = -3*±i. 5 -4| By means of the condition given in Th. 1, Cor. 1, it may be shown even more easily that 5 — V— 5, 11 — 4V — 5 is not a basis of (3, 1 + V— 5) ; for 1, V — 5 being a basis of the realm, we have 3 = 3-1+0- y/~=5, 5 — V^5 = 5 • I + — I • V 7 ^, 1 + V^5=I • 1 + 1 • V" 37 ! 11— 4V T77 5 = ii • i+ — 4- V^, 5 11 -4 Ex. 2. We can show in like manner that ■3 + V 3 t 3 + 5V — 3 2 is not a b asis of the ideal (— 2 + »,' — 1 + 5&O of the realm &(V — 3). 1, * ~r V 3 b e i n g taken as a basis of the realm. 2 Proceeding as in Ex. 1 we see that the necessary and sufficient con- dition for — 2 + «, — 1 4- 5W to be a basis of the given ideal is that the equation (d + <*»*) (— 2 -f- «) -f- (c 2 + &«) (_ 1 4. 5 co) 14) = / 1 (-2 + o,)+/ 2 (-I + 5«) shall be satisfied by rational integral values of h, h for every possible choice of c u di, c 2 , d 2 . Performing the multiplications indicated in 14), putting w"= — 1 + w , and equating coefficients of like powers of « in the two members, we have the equations — 2Ci — C 2 -\- di — 5rf 2 s= — 2/1 — h, which give These equations evidently do not give integral values for h, U for every possible choice of Ci, di, c 2 , d 2 ', for example, for a = d x = c 2 = d 2 = 1. Hence — 2 -f- w, — 1 -f- 5 W is not a basis of ( — 2 -)- w, — i_[_5o>). We have chosen an ideal of the realm k{\/ — 3), in which the unique fac- torization law holds in the ordinary sense, to emphasize the fact that with the introduction of ideals all quadratic realms are to be treated 1 + 5^2 — 3^i- -6d 2 - ~-h + 5k — 9^1 + 2fl?x- -31&2 = — 9k 9c 2 — 5^i- - I7c? 2 - = 9/2, THE IDEALS OF A QUADRATIC REALM. 30I alike, and that all theorems to be deduced hereafter will be equally valid whether the unique factorization law holds in the ordinary sense or not. Ex. 3. Show bot h by the above method and by the nature of a canonical basis that 7, 3 + V~ 5 is a basis of the ideal (7, 3 + V =r 5) ; also that 3 + V— 5, 5 + 4V — 5 is a basis of the same ideal. In k(yj^23) show that 3, I + > ^~ 23 is a basis of the ideal (3, * + ^~ 2 A . a i so q _1_ ^-v / 2 "3 that 4 + V — 2 3, — is a basis of the same ideal. In k{yJ6) showthat 10 + $\/6, 6-\-2\J6 is a basis of the ideal (10 + 3V6, 6 + 2V6). Ex. 4. Show that 7 -f 7 V— 5, — 5 + 3V— 5 is not a basis of the ideal (7 + 7V— 5, — 5 + 3V— 5). _ Ex. 5. Show that (3* — J is a principal ideal of &(V~i3)- Show that the two ideals (2, I + ^~ I5 ) and (3, ~ + ^~ I5 ) are both non-principal ideals of k(yj — 15), but that their product is a prin- cipal _JdeaL Show that (2, 1 + V — 13) is a non-principal ideal of *(V— 13). § 3. Conjugate of an Ideal. // a be any ideal, the ideal, zvhose numbers are the conjugates of the numbers of a, is called the conjugate of a and is denoted by a'. 1 It is easily seen that, if a= (a lt a 2 , • • •, a„) be any ideal, then a' = (a/, a/, •••', a/) is the conjugate of a; for, if ^i a i + A 2 a 2 + * " ' + ^nCln be any number of a, its conjugate A/a/ + A/a/ H h A»'a»' is a number of a, and vice versa. Moreover, if a x w x -f- a 2 o> 2 , fr^ + 6 2 w 2 be a basis of a, where * ti o> 2 is a basis of the realm, then a^/ + « 2 ° ) 2 / > ^i w i' + ^^ is a basis of a'. The truth of the last statement is readily seen when we remember that, if a 1 (o 1 + a 2 w 2 , fr^ -f- b 2 a> 2 be a basis of a, then every number, a, of a can be expressed in the form a==a(a 1 oi 1 -}-a 2 a> 2 ) -f- b{b 1 (a 1 + & 2 <o 2 ), where a and & are rational integers. The corresponding number, a', of a', being expressible in the form a , = a(a 1 (i) l ' + a 2 o>/) + b^b^ + b 2 o) 2 ), it is evident that a^/ + a 2 w/, &!<»/ + & 2 <o/ is a basis of a'. 1 Hilbert: Bericht, p. 191. 302 THE IDEALS OF A QUADRATIC REALM. For example, the conjugate of (2 + 3 V — 5, 7 + 2 V — 5> *7) is (2 — 3V zr 5, 7^2y^5, 17)^ also since 3, 1 +V^5 is a basis of (3, 14.y-~.-5), 3,1— V :=r 5 is a basis of (3> x — V— 5)- § 4. Equality of Ideals. Two ideals, a,= (a 1 ,a 2 ,---,a r ), and &,= (/?i,/? 2 > ••-,/?«), are said to be equal, and we write a = B, when every number of a is a number of fc and every number of fc is a number of a. The necessary and sufficient condition for the equality of a and h is that every number, ai, defining a shall be expressible in the form ai = A 1 1 + A 2 flH hA 8 /?s, and that every number, /?y, defining b shall be expressible in the form pj = fijO^ + fi 2 a 2 -\ f- {x r a r . The practical test of equality is to see whether the symbol defining either one of the ideals can be reduced to that defining the other by the introduction and omission of numbers under the laws given in the preceding paragraph. 1 Ex. 1. Show that (6 + 2V — 5, 56 + 7V~ 5) = 05 + 5V — 5, 14). Ex. 2. Show that ( I+ 2 >/l3 , 5+8/^3, 5 + 2 fft$) = (5 + 14 1A3, 6/IE3). Ex. 3. Show that (7, 1 +V=i3) 4= (7, 1— V=i3). § 5. Multiplication of Ideals. By the product ah of the two ideals w understood the ideal, whose numbers consist of all possible products of a number of a by a number of b, together with all linear combinations of such products with coefficients which are any integers of the realm. 2 We have therefore ah = (aA, • • •, a^, • -,OrA, • • •, a r p 8 ), 'See Chap. VIII, §9. 2 Hilbert: Bericht, p. 183; also see Chap. VIII, § 11. THE IDEALS OF A QUADRATIC REALM. 303 where the numbers defining qB are all possible products of the numbers defining a by those defining b. If a=(a) and 5= (ft, ft, —,ft), then oB = (aft, aft, • • •, aft). If a=(a) and B=(0), then ab=(a£), and we see that the product of two principal ideals is a principal ideal. It is evident from the definition that ah = ba, and that ctb-c = a-bc; that is, that the commutative and associative laws of multiplica- tion hold for ideals. Ex. Show that (2, V^6) (3, i - V^6) (5, 2 + V^26) = (2 + V -26). § 6. Divisibility of Ideals. The Unit Ideal. Prime Ideals. An ideal, a, is said to be divisible by an ideal, b, when there exists an ideal, c, such that a = bc. We say that b and c are divisors of o, and that a is a multiple of b and c. We have as a direct consequence of the above definition : // each of a series of ideals a lt ct 2 , a 3 , •••, be a multiple of the next following one, then each is a multiple of all that follow. If two or more ideals, a, b, c, •••, be each divisible by an ideal j, j is said to be a common divisor or common factor of a, b, c, • • •. Theorem 2. // the ideal a be divisible by the ideal b, then all numbers of a belong to b. For suppose that a = bc, 304 THE IDEALS OF A QUADRATIC REALM. where a=(a p a 2 ,"-,a r ), b= (ft,ft, ••-,/?*), c= (yi,y 8 , •••,?#); then a=(ftyi, "-fPxyt, •■•,fty„ •••jftyt)- The numbers, fty 15 ■•■,(3 s yt, defining a are seen to be numbers of ft. Hence all numbers of a are numbers of B. Therefore h = (ft, ft, ••-,/?*, a if a 2 , • • •, a r ), and c== (yi,y 2> "$yu <h><**> •••,ot r ). Cor. 1. // fcco ideals fo .swc/j fto £?ac/i w a divisor of the other, they are identical. The converse of Theorem 2 is also true ; that is, if all numbers of a be numbers of b, a is divisible by h, but its proof must be deferred until some necessary theorems have been demonstrated. Every ideal is divisible by the ideal (1), which consists of all integers of the realm. Therefore (1) is called the unit ideal. The only ideal having this property is evidently (1), for every divisor of (1) contains all integers of the realm and is (1). We observe that (r/) = (1), where 77 is any unit of the realm. Since (i)ct=a, there is, in the case of ideals, no distinction to be made corresponding to that made between associated integers. An ideal, not the unit ideal and divisible only by itself and the unit ideal, is called a prime ideal. In k{ V=5), (2, 1 +V^), (3, 1 +V :: ^5), (3. 1 -V— S) were shown to be prime ideals (see p. 264). Two ideals are said to be prime to each other when they have no common divisor except (1). Two integers a and fS of the realm are said to be prime to each other when the principal ideals (a) and (ft) are prime to each other. For the sake of brevity we shall often say that an integer a is divisible by an ideal a, instead of saying that the principal ideal (a) is divisible by a. The latter meaning is, of course, always to be understood. Similar expressions, such as " a prime to a," " the greatest common divisor of a and a," etc., are to be taken in the same sense. THE IDEALS OF A QUADRATIC REALM. 305 By means of the definition of divisibility and the fact that every ideal has a basis, we can prove the following important theorem : Theorem 3. An ideal j is divisible by only a finite number of different ideals. 1 Let a = (ao) 1 + b<a 2 , cw 1 + d<a 2 ) be a divisor of j, where aoi 1 -f- bo) 2 , co) 1 -f du> 2 is a basis of a, <o lt <o 2 being a basis of the realm. Let /? be any number of j. Then, since n [/?]=/?/?' = 0, mod i, and a is a divisor of }, we see that by Th. 2 n[fi] e=o, mod a; that is, the rational integer n[(3] belongs to every divisor of \. Denote now w[/3] by n and let a lf b x , c lt d 1 be the smallest posi- tive remainders of a, b, c, d with respect to n. Then a = (aio-L -f- b(o 2 , cui x -\- do) 2 , n) =3 {a 1 o> 1 -f- b x u) 2 , c^-^ -\- d^^ n) 1) Suppose every divisor of j to be expressed in the form 1). Since a lt b lf c ls d x can each take only the finite number of values o, 1,2, •••, \n\ — 1, it is evident that the number of different divisors of \ is finite. § 7. Unique Factorization Theorem for Ideals. We shall now proceed to prove the theorem whose truth is the raison d'etre of the ideal; that is, that every ideal can be repre- sented in one and only one way as a product of prime ideals. This theorem will enable us to develop for the ideals of the general quadratic realm a series of theorems similar to those already given for the integers of certain realms in which the ordinary unique factorization theorem held. The proof of the unique factorization theorem for the ideals 1 Hilbert: Bericht, Hiilfsatz 1. 20 306 THE IDEALS OF A QUADRATIC REALM. of the general quadratic realm will be very like that for the integers of R, &(V — i), k(\/ — 3) an d &(V 2 )- It depends directly upon the theorem that, if the product of two ideals be divisible by a prime ideal, at least one of the factors must be divisible by this prime ideal. The latter theorem is a consequence of a series of three theorems which have no analogues in those relating to integers. It depends, in the first place, directly upon the theorem referred to on p. 304, that, if all the numbers of an ideal belong to another ideal, the first ideal is divisible by the second. This depends, in turn, upon the theorem, that, if the products ab, etc of two ideals, b and c, by a third ideal a be equal, then b = c, and this upon the theorem, that for every ideal there exists another ideal such that the product of the two is a principal ideal. This last theorem is the starting point of the proof of the unique factorization theorem and needs for its demonstration a theorem which we shall proceed to give. Theorem 4. // the coefficients, a lt a 2 , J3 1} /? 2 of the two ra- tional integral functions of x, 4>{x) ==a 1 x-\-a 2 and \f/(x) =/3 ± x + /? 2 , be integers of k(\/m) and w, an integer of k(\/m), divide each of the coefficients, y 19 y 2 , y 3 , of the product of the two functions, F (x) = <f>(x) ^(x) = a^x* + ( fll ft + a£ x )x + a 2 (3 2 = 71^ + 72^ + 73, then each of the numbers a^ ly a^ 2 , a 2 f3 1} a x p 2 is divisible by w. 1 Suppose a x and /?! =|= o. Then y 1 =(= o. We have Hence — a 2 /?i/7i an d — ^A/Vi are tne roots of 1 Hurwitz : Nachr. der K. Ges. der Wiss. zu Gottingen, 1895 ; also Hil- bert: Bericht, Hiilfsatz 2. THE IDEALS OF A QUADRATIC REALM. 307 Let £ represent either a 2 p x or a 1 fS 2 ; we have V vj 7i\ rj % and, multiplying this equation by y x 2 , €°~ — y 2 £ + 7i7 3 — o. Since y 2 and yiH are divisible by w and o> 2 respectively, the coefficients of the equation i-Y-H 1 ) \(0 ) to \ to J + O) 2 °' that £/o> satisfies, are integers. Hence £/« is an integer (Chap. IX, Th. 9, Cor. 1 ) ; that is, a x /3 2 and a^ are divisible by ». Theorem 5. For ^z/^ry icfea/ a of a quadratic realm there exists an ideal h of the realm such that the product ah is a prin- cipal ideal. 1 Let a= (a lt a 2 ) where a lf a 2 is a basis 2 of a. We shall show that the conjugate of a, that is, the ideal &,= (a 1 ', a 2 '), where a/, a 2 ' are the conjugates of a lf a 2 , has the desired property. 3 Let <f>(x)=a 1 x -{-a 2 and ij/(x) = a 1 , x + a 2 '. Form the product 4>(x)iKx) = a 1 a 1 , x 2 -f- (a^a/ +a 1 , a 2 ) jir H-c^A/ = 7i^ 2 + y2^ + y 3 - Let be a number defining the realm and let the irreducible rational equation of which 6 is a root be x 2 -\- a r r -f a 2 = o. 1 ) Since y u y 2 , y 3 are symmetric functions of the roots of 1), they 'Hilbert: Bericht, Satz 8. " The simplification effected by the use of the basis representation of an ideal is that, in a quadratic realm, the basis consists of two numbers and hence Th. 4 need be proved only for functions of the first degree. 3 In the realm of the nth degree the ideal that will have the desired property is the product of the conjugates of a. This ideal is, however, not the only ideal having the desired property (Chap. XIV, § 1). 3<d8 the ideals of a quadratic realm. are rational integral functions of its coefficients i, a lt a 2 . Hence Yi> 72j 73 are rational numbers. But y lt y 2 , y 3 are also integers, since a lf a/, a 2 , a 2 are integers (Chap. IX, Th. 8, Cor. 2). Hence y u y 2 , y 3 are rational integers. Let a be the greatest common divisor of y lf y 2 , y 3 . Then ah=(a 1 a 1 ', a,a 2 ', a t f a 2 , a 2 a 2 ') is equal to the principal ideal (a) ; for by Chap. II, Th. B, we can find three rational integers, f w t 2 , t 3 , such that a = ^i7i + * 2 y 2 + hys = f^a/ + ^(o^a,/ + a/a 2 ) + t z a 2 a 2 ' Hence a is a number of ah and we have ah = (a 1 a 1 ' t a x a 2 , a^a 2 , a 2 a 2 , a). But by Th. 4 each of the numbers a^', a ± a 2 , a^a 2 , a 2 a 2 is a multiple of a. Hence we can omit them from the symbol and have ah=(a). Therefore b is the required ideal. It will be observed that we have proved that the product of an ideal of a quadratic realm by its conjugate is a rational principal ideal. This will be of use later. Theorem 6. // a, h and c be ideals and ac = 6c, then a^b. 1 Let a=(a 1 ,a 2 ,--,a r ), h= (ft, ft, ••-,&), c= (yi,y 2 , •••,yt), and let m, = (fi lt fi 2 , --,iin), be an ideal such that cm=(y 1 ^ 1 , -,y</in) = (a), a principal ideal. Then ocm = km, or a (a)=h(a), or (a x a, a 2 a, • • •, a r a) =(fta, fta, ■ • •, fta). Since these two ideals are equal, every number of the one must 1 Hilbert : Bericht, Satz 9. THE IDEALS OF A QUADRATIC REALM. 309 be a linear combination of the numbers defining the other, with coefficients which are integers of the realm. Hence, if aia be any number of the first and PjO. any number of the second, we have aid = ^p x a + £ 2 p 2 a + • • • + ZsPsCL, and pjOL = rj^OL^ -f~ f] 2 CL 2 0. + • • • -f- TfrCLfOL, where the £'s and rfs are integers of the realm. Hence a«=!a& + *AH \-i*fr, Pj = n^ + r) 2 a 2 -j [- ^a,.. Hence every number of a is a number of b, and every number of h is a number of a, and consequently a = 6. Theorem 7. 7/ a// numbers of an ideal c belong to an ideal a, c is divisible by a. 1 Let c= (y u ■■-,yt) and a= (a lf •••,a r ,y 1 , •••,y*) ; and let m,= (f^, ••-,/*«), be an ideal such that ftm=(o 1 /H, • • • , a r ^n, yi/*!, •••,y*/*«) = (a), a principal ideal. Then all numbers of am, and hence Yifi 1 ,'",yil**,'"tytiHf'"t ytfin, must be divisible by a. Hence all numbers of rcic= (yi/^i, •••>yiMn, •••>y*A*n) are divisible by a ; that is, mc = (v x a, • • • , v n *a) = (a) (v x , • • ■ , v„« ) = (a)b. 2) Multiplying both members of 2) by a, we have cmtc= (a)o6, or c = ah. Hence c is divisible by a. 1 Hilbert: Bericht, Satz 10. 3IO THE IDEALS OF A QUADRATIC REALM. This theorem justifies our use of the notation a = o, mod a, to denote that (a) is a multiple of a. For, if a be a number of a, then from the above theorem it follows that (a) is divisible by a. From Th. 2 we saw that a necessary condition for an ideal a to be divisible by an ideal b is that all numbers of a shall belong to b; from Th. 7 we see that this condition is also sufficient. Hence every common divisor, b,= (8 lf •■■,8t), of two ideals a=(a 1 , •••,Or), &=(&, •••,£,) must contain all numbers of both a and b ; that is, where 8 X , •••,$* are any integers of the realm, and every ideal of this form is a common divisor of a and b. Among the common divisors of a and b is one, g, to which belong no numbers other than the numbers of a and b, together with the linear combinations of these numbers; that is, This ideal g is divisible by every common divisor, b, of a and b, for b must contain all numbers of a and b, and hence all the numbers of g, and therefore is a divisor of g. As in the case of rational integers, g is called the greatest common divisor of a and b. That g is the only ideal having this property is evident ; for did a second, f), exist, then g must be divisible by I) and I) by g, and hence g and § be identical (Th. 2, Cor.). Likewise the necessary and sufficient condition that an ideal, tn, shall be a common multiple of a and b is that all numbers of m shall be common to both a and b. Among the common multiples of a and b is one to which belong all numbers common to both a and b, together with the linear combinations of these numbers. This ideal, I, is evidently a divisor of every common multiple THE IDEALS OF A QUADRATIC REALM. 3 1 I of a and b. That I, moreover, is the only ideal having this prop- erty may be shown as in the case of g. As in the case of rational integers, I is called the least common multiple of a and b. We shall denote the greatest common divisor of a and b by the symbol a + b, and the least common multiple of a and b by the symbol a — b. No idea of addition or subtraction is to be conveyed by these symbols. From Theorems 2 and 7 we have the important result that an ideal a,= (a 1 ,O a , •••,a r ), is the greatest common divisor of the numbers defining it considered as principal ideals; that is, a is the greatest common divisor of (o^), (a 2 ), •••, (a r ). The fact that we can at once write the greatest common divisor of any number of ideals by placing in a single symbol all the numbers defining the ideals is of use in numerical work with ideals. Thus, if we can show that the greatest common divisor of two ideals so written is (1), we know that the ideals are prime to each other. Ex. The greatest common divisor of (3 -f- V — 5) and (8-f-V — 5) is (3 + V — 5> 8 + V — 5), and we have (3 + V^5, 8 + V^l) = (3 + V"= r 5, 8+V=5, 5, 14) = (3 + V =: 5, 8 + V=I), 5, 14, = 0) Hence (3 -+- V — 5) and (8+V — 5) are prime to each other. The ideas of the greatest common divisor and least common multiple of two ideals may be at once extended to any number of ideals. Thus, if a lt a 2 , • • •, a m be any number of ideals of a realm, there is among the common divisors of a^cio, ■•■,a m one, g, to which belong no numbers other than the numbers of a lf a 2 , •••, Qm, together with the linear combinations of these numbers; that is, if a 1 =(a 1 ,-'-,ar), 0*= (ft, •••,/*• ),•••, fc»=G«i> •••»**#), then g= (a lt •••,a r ,^ 1 , ••-,/?*, ••-,/*!, ••-,!»#). That g is divisible by every common divisor of a v a 2 , •••,a m and is the only ideal having this property is seen as in the case of two ideals. We call g the greatest common divisor of a lf a 2 , • • • , a m . 312 THE IDEALS OF A QUADRATIC REALM. Likewise the ideal, I, to which belong all numbers common to a 19 a 2 , --'yOm, together with their linear combinations and no others, is evidently the only common multiple of a : , a 2 , • • •, a m that is a divisor of every common multiple of a lf a 2 , •••, a m . It is therefore called the least common multiple of a lf a 2i •••,o m . We write symbolically Q = a 1 + a 2 -\ \-0m t and l = a ± — a 2 — ••• — dm- We have as an immediate consequence of Th. 7 and the defini- tion of the least common multiple of two or more ideals the following : Cor. // an ideal a be divisible by each of the ideals h lt b 2 , •••, B r , then a is divisible by the least common multiple of h t , b 2 , ■ ••, b r . We shall see later that the greatest common divisor, as defined above for ideals, possesses the remaining two properties which distinguished the greatest common divisor of two or more integers in those realms in which the unique factorization law held in the ordinary sense (see p. 318). We have now a full justification of our introduction in feCV 31 ^) of the ideals ( 2 ^_! +V zr 5), (3, i+V^), ( 2 > 1 — V — 5) and (3, 1 — V — 5) as the greatest common divisors respectively of (2) and (i+V — 5)» (3) anc * (1 +V — 5), (2) and (1— V zr 5),and (3) and (i— -yCTj)- Th. 7 having been proved, the remaining theorems necessary for the proof of the unique factorization theorem and the proof of that theorem itself for ideals are strictly analogous to the cor- responding theorems in the realms in which the unique factoriza- tion law held in the ordinary sense. It may seem singular that the divisors of an ideal, a, are in a way larger systems of numbers than the ideal, a, itself; that is, they contain not only the numbers of a but in addition any other numbers of the realm that we choose to introduce. When, however, we remember that by Th. 7 an ideal divides every one of its numbers considered as a principal ideal, it is evident that, in THE IDEALS OF A QUADRATIC REALM. 313 general, the more numbers we introduce into the symbol of an ideal, that are not linear combinations of those already there, so much the more do we narrow the ideal by thus placing more restrictions upon it. For example; the ideal (14, 3 -f V — 14) is the greatest common di- visor of (14) and (3 + V — 1 4)» and the ideal (14, 3 + yJ~ZT^ 2 ), that contains all numbers of (14, 3 + V — 14) and other numbers be- sides, divides not o nly (14) and (3 -f- >/ -^4), and hence is a divisor of (14, 3 + V — J 4)» DUt must also divide (2). It is analogous to the case of rational integers when we observe that 120 is divisible by every common divisor of 120 and 18, and that every common divisor of 120 and 18 is divisible by the common divisors of 120, 18 and 4. Examples. 1. Find the greatest common divisor of (8 + V — 14) and (4— V— 14). 2. Find the greatest common divisor of (26, 10-J-2V — 14, 13 V— 14, — 1 4+5 V— 14) and (10, 2 + 2V— 14, 5V— 14, — 14+V— 14). 3. Show that the two ideals (5, — 4+V-—14) and (13, 5 — 12 V — 14) are prime to each other. 4. Making use of form of canonical basis, show that (23, 8 -f- V — 5) is a prime ideal. 5. Show that (p, b -\-o)) is a prime ideal, p being a rational prime, b any rational integer, and 1, w a basis of the realm. 6. If p and q be two different rational primes, show that in no realm can (p) and (q) have a common ideal factor different from (1). 7. Show that (1 +V — 5) is tne l east common multiple of (3, 1 +V = S) and (2, 1 -f-V^)- 8. Find the least common multiple of (6, 4+V — 1 4) and (10, 6+V— 14)- 9. Show that, if a be divisible by a x and 6 by 6 X , then ab is divisible by a^. 10. Show that, if ah be divisible by QC, then b is divisible by c and in particular that, if a be divisible by ab, then b= (1). 11. Show that, if a, B and c be any ideals, then [a + B]c=oc + Bc. 314 THE IDEALS OF A QUADRATIC REALM. 12. Show that [a + B + c] [Bc + ca + aB] = [B + c][c + a][a + B]. 13. Show that, if a be divisible by a x , and ft by h x , then a + B is divisible by a x + Bi, and also that a — B is divisible by a x — B r 14. Show that, if a and B be any two ideals, then a + B is the system of all numbers of the form a -\- ft where a is a number of a and j3 a number of B. 15. Show that, if a, B and c be any three ideals, a — [B — c] == [a — B] — c. 16. Show that [a + B][a — 6]=a6. 17. Show that, if a and B be prime to each other, then a — B := aB. Theorem 8. // a and B be any two ideals prime to each other, there exist a number a of a and a number |8o/B such that a + ^1. 1 Let a = (a lt a 2 , • • •, a r ) and B = (ft, ft, • • •, ft) . Since a and B are prime to each other their greatest common divisor is ( 1 ) ; that is, a + B = (Op a 2 , • • •, Or, ft, ft, • • •, ft) = ( 1). But, since 1 is a number of a + B, it must be a linear combination Of CLi,a 2 y ••*»«r, ft, ft, "'yps', that is, gA + ^ 2 0t 2 H h £rCL r + l?ift + ^ 2 ft H h >7*^. = I, where the |'s and ^'s are integers of the realm. But £ x a x + | 2 a 2 + • • • + | r a r = a, is a number of a, and r) x fi x + ?7 2 ft + • • • + ^sft = ft is a number of B, and we have x This is the analogue of Th. B. See Hilbert: Bericht, Satz 11. THE IDEALS OF A QUADRATIC REALM. 3 I 5. Cor. // a lf ct 2 , • • •, a m be ideals whose greatest common divisor is (/), then there exist in c^, ct 2 , •••, a w numbers a lf a 2 , •••,a fft , respectively, such that <2i + a 2 H \-a m = i. Theorem 9. // the product of two ideals, a and b, be divis- ible by a prime ideal p, at least one of the ideals is divisible by p. 1 Assume that a is not divisible by p. Then a and p are prime to each other and there exists by Th. 8 a number, a, of a and a number, ?r, of p such that a-\-7r= 1. Let now /? be any number of b, and multiply the last equation by /?; then But aft is a number of ab, and hence by Th. 2 of p, since ab is divisible by p. Moreover, irfi is a number of p. Hence ft is a number of £ ; that is, all numbers of b are numbers of p, and b is therefore by Th. 7 divisible by p. Cor. 1. // ffo product of any number of ideals be divisible by a prime ideal, p, at least one of the ideals is divisible by p. Cor. 2. // neither of two ideals be divisible by a prime ideal, p, their product is not divisible by p. Cor. 3. // the product of two ideals, a and b, be divisible by an ideal, j, and neither a nor b be divisible by j, then \ is a com- posite ideal. If all the ideals of a realm be principal ideals, the unique fac- torization theorem in the usual form holds for the integers of the realm; for, if a and (3 be any two integers prime to each other in the usual sense, then the ideals (a) and (/?) are prime to each other, for all factors of (a) and (/?) are principal ideals. Hence the ideal (a, /?) must be the unit ideal (1) ; for (a, /?) divides both (a) and (/?) and they have no common divisor other than (1). Since (a, £) = (l), 'This is the analogue of Th. C. See Hilbert: Bericht, Satz 11. 3 16 THE IDEALS OF A QUADRATIC REALM. there must exist two integers, | and r), of the realm such that Th. B would therefore hold for the integers of the realm, and we have seen that Th. C, and hence the unique factorization theorem, follow immediately. The converse of this, that, when- ever the unique factorization theorem in its usual form holds for the integers of a realm, the ideals of the realm are all prin- cipal ideals, is evident; for, if a,= (a lf a 2 ,---,a r ), be any ideal, the numbers a lt a 2 , • • • , a r have a greatest common divisor 8, and since the unique factorization law holds for the integers of the realm, we can find integers (Chap. V, Th. B, Cor. 2) £ lt £ 2 , --,£ r , such that aJi + a 2 £ 2 H h a-r£r = 8. Hence we have a = (a 19 a 2 , ~-,a r ) = (a 15 a a , -, a r , 8) = (8), a principal ideal. Theorem 10. Every ideal can be represented in one and only one way as the product of prime ideals. 1 Let j be any ideal. If \ be a prime ideal the theorem is evident. If j be not a prime ideal, it has some divisor, a, different from j and from (1). Then i=o&. If a be not a prime ideal we have a = a x a 2 , where a x and a 2 are both different from a and ( 1 ) . Then If any of the ideals a lt a 2 , 6 be not prime, we factor them, and, proceeding in this manner, we reach finally a point where the factorization can be carried no further, for an ideal, j, is divisible by only a finite number of ideals (Th. 3). The ideal j has now been resolved into its prime ideal factors. 1 Hilbert : Bericht, Satz 7. THE IDEALS OF A QUADRATIC REALM. 317 Let i=M 2 '-'Pr, where p 1} p 2 ,---,p r are prime ideals, be the representation so obtained. We shall show that this representation is unique. Suppose that j could be represented in another way as a product of prime ideals, say Then pfa • . . p r = q x q z . - . q # . 3) Since p t is a divisor of the product q t q 2 • • • q 8 , it is a divisor of one of its factors (Th. 9, Cor. 1), say q lf from which follows *>i=qi- Then it follows from 3) that p 2 ...p r = q 2 ••• q a . Proceeding in this manner, we see that for each factor in the product pjp % • • • p r there is an equal one in the product q^ 2 • • • q 8 , and, reversing the process, that for each factor in the product q t q 2 '" <\s, there is an equal one in the product p x p 2 • • • p r , and that, if a factor be repeated in one product, it is repeated exactly as often in the other. The two representations are therefore identical, and the theorem is proved. Cor. If the product of two ideals, a, b, be divisible by an ideal, m, and a be prime to m, then fc is divisible by m. If we denote by p lt p 2 ,'",pr the different prime ideals that are factors of j, and by c x ,e 2 ,---,e r the number of times that they are repeated respectively, we have It is convenient sometimes to allow one or more of the expo- nents to take the value o, a = o indicating that j does not contain pi as a factor. It is evident that an ideal j is then and only then divisible by an ideal b if every prime ideal which divides b occurs to at least as high a power as a factor in j as it does in b. Every divisor of j has therefore the form b = p 1 m >V 2 m *'-'p r mr , 4) 3 l8 THE IDEALS OF A QUADRATIC REALM. where nti^ei; i=i,2,--,r J and every ideal of the form 4) is a divisor of j. If we let m* run through the *< + x values, o, I, ••-, e iy and do this for each of the exponents m ls m 2 , •••, m r , we obtain different sets of values for these exponents, and each of these sets gives a different divisor of j. The number of divisors of j is therefore (e 1 + 1) (e 2 -f- 1) ••• (e r -\- 1). If i = p 1 mi ^2 m2 •••^r mr , and f) = p^/ 2 • • • p r nr , where p ls p 2 , •••, p r are different prime ideals, be any two ideals, the ideal where gi is the lesser of the two exponents mi and m(i = l, 2, •••,r), is the greatest common divisor of j and f). The ideal where U is the greater of the two exponents mi and «<(t=l, 2, •••,r) is the least common multiple of j and f). We see from this representation of the greatest common divisor, g, of j and § that, of all common divisors of } and % g has the greatest norm, and that the quotients, j/g and Ij/g, are prime to each other (see p. 18). Theorem ii. If a. and m be any two ideals, there exists a number, a, of a such that the quotient (a) /a is prime to m. For example, if a,= (2, 1 -f-V — 5), and m,= (3, 1 +V — 5), be the given ideals, then a = 2 satisfies the requirements of the theorem, for THE IDEALS OF A QUADRATIC REALM. 319 that is easily seen to be prime to (3, 1 -)-y — 5). If a , = (2, i+V^5), and m,= (1 +V =I 5), be the given ideals, then a, = 2 + 1 +V ir 5, = 3 +V =r 5, satisfies the re- quirements of the theorem for (3 + v^ , 3 + v ~ s) (2,i+v=T) (7 ' 3 + v sh that is prime to (1 +V — 5). For the actual determination of a in general see Chap. XII, § 7. We proceed now to prove the theorem. The truth of this theorem for the case where m is any prime ideal p is at once evident. For, if there did not exist a number, a of a such that (a)/a is not divisible by p, then all numbers of a would belong to ap and by Th. 7 a would be divisible by ap, which is impossible. To prove the theorem for the general case, let the different prime factors of m be p lt p 2 , •••,p mi and form the products <*! = ap 2 • • • p m , a 2 = ap!p 3 • • • p m , • • • , a m = ap x • • • p m - x , which consist of a multiplied in turn by the combinations of Pi>p2> ••-,Pm taken m — 1 at a time. Let a lt a 2 , -',a m be num- bers of a lt a 2 , •••, a OT respectively, such that (aO/c^, (« 2 )/a 2 , • • •, (a m )/a m are prime respectively to p lt p 2 , • • •, p m , the existence of such numbers having been proved above since p lf p 2 , •••, p m are prime ideals. Then a = a 1 -\-a 2 -{ \-a m , is the required number; for a is divisible by a, since a x ,a 2 , -",a m are all divisible by a, cu being divisible by a, whence all numbers of cu belong to a; moreover, a is not divisible by any of the m products api, ap 2 , '-,ap m , as, for example, ap lt since a 2 ,a s , •••,a m are all divisible by ap lf but a x is not divisible by ap x . It is evident, therefore, that the quotient (a) /a is divisible by 320 THE IDEALS OF A QUADRATIC REALM. none of the prime factors p lf p 2 , ••• i p m of m, and hence is prime to m. Hence a is the required number. Theorem 12. In every ideal, a, there exist two numbers, a lf a 2 , such that a=(a 1 ,a 2 ); that is, such that a is the greatest common divisor of (a ± ) and (a 2 ). Let a t be any number of a. By Th. 11 there exists in a a number, a 2 , such that the quotient {a 2 )/a is prime to (a x ) ; or, in other words, such that the greatest common divisor of (a x ) and (a 2 ) is a. But, since a is the greatest common divisor of (a t ) and (a 2 ), it contains all and only numbers of the form P& + p 2 a 2 , where f} v fi 2 are any integers of the realm. Hence a=(a lt a 2 ). The truth of this theorem is at once evident for quadratic realms for we have shown (Th. 1) the existence in every ideal, a, of a quadratic realm of two numbers ha h such that a= («i, t 2 ). The proof in the above form has been given, however, as it applies to the general realm of the Mth degree; see Hilbert : Bericht, Satz 12. The following theorem is given not only for its own interest but because from it we obtain a new proof of Th. 11 that is not dependent upon the unique factorization theorem. Dedekind makes the unique factorization theorem depend upon Th. 13 (see Dirichlet-Dedekind, § 178, IX). Theorem 13. // the ideal a be divisible by none of the ideals Ci» c 2 , •••,(:„, then there is a number, a, of a that is contained in none of the ideals c lf c 2 , ••-•, c«. If a should be a principal ideal, the theorem is evident. Also, if there should be only a single ideal, c, the theorem holds, for, if all numbers of a were divisible by c, a would be divisible by c, THE IDEALS OF A QUADRATIC REALM. 32 1 which is contrary to the original hypothesis. We shall now prove that, if the theorem hold for n < r it holds for n = r, and hence, since it is true for n=i, it holds in general. To each of the ideals c lf c 2 , •••, c r , as c«, there corresponds an ideal h 8 such that ab 8 = a — c„ where h 8 is evidently different from (i). The ideal a is divisible by none of the r products db lt ah 2 , • • •, ab r , 5) since all of the b's are different from (1). But each one of the c's divides one of these products. Hence, if we can prove the existence of a number of a, which belongs to none of the products 5), this number will be the desired number a, for if a were divisible by c«, it, being divisible by a, would be divisible by the least common multiple of a and c«; that is, ab 8 . We have now two cases to consider according as the ideals h lf 6 2 , •••, fc r are, or are not, prime each to each. If they be not prime each to each, some pair of them, say h lf B 2 , must have a greatest common divisor, h t + ft 2 , that is different from ( 1 ) . Then a is not divisible by a(B x + b 2 ), and hence, according to our assumption that the theorem holds for n < r, there exists in a a number, a, that is divisible by none of the r — 1 ideals a(B 1 + B 2 ),ab 3 , --,cib r , and hence also is not divisible by ab 1 and ab 2 , since they are divis- ible by a(B x -|-B 2 ). Therefore a is not divisible by any of the c's. We must consider now the case where the r ideals, b lt b 2 , • • •, B r are prime each to each. Each of these ideals, as 6„ is prime to the product, $«, of all the remaining ones, and, since they are all different from (1), Ij s is not divisible by &,. Hence at} 8 is not divisible by ab 8 , and there is therefore a number a 8 , in a$ 8 that is not divisible by ab 8 . The number a, = a t + a 2 -\ \-a r , where a lf a 2 , • • •, a r are numbers of a^ lf at) 2 , • • •, df) r respectively, is a number of a, for each 21 322 THE IDEALS OF A QUADRATIC REALM. of the numbers a lf a 2> -",a r is a number of an ideal divisible by a, and is therefore a number of a. Moreover, a is divisible by none of the r products ab lf ab 2 , • • • , ah r ; for, since the ideals § 2 , § 8 , • • •, § r are all divisible by i) lf all the ideals ai) 2 , •••,af) r are divisible by ab 1} and hence a 2 , a 3 , ---jCtr are numbers of qBj. But a x is not a number of ab x , and hence a is not a number of abj. In like manner it may be proved that a is divisible by none of the ideals ab 2 , ab s , • • •, ab r . Hence a is the number sought. Second Proof of Theorem II. 1 If m=(i), every number of a satisfies the requirement of Th. ii. If m+C 1 )* let c,, c 2 , •••,c, l be all the ideals different from a that divide am and are divisible by a. By Th. 3 these ideals are finite in number and hence there is in a a number, a, that is divisible by none of them (Th. 13). Hence the greatest common divisor, am + (a), of am and (a) is different from all the c's. But am -f (a) divides am and is divisible by a, and the only ideal different from the c's, that has this property, is a. Hence am-\-(a)=a, 6) or, what is the same thing, (a) /a is prime to m. From 6) it follows at once that am — (a) =m(a). 1 Dirichlet-Dedekind : §178, X. CHAPTER XII. Congruences whose Moduli are Ideals. 1 § i. Definition. Elementary Theorems. If the difference of two integers, a and /?, be a number of the ideal o, we have said that a was congruent to /? with respect to the modulus a, and have denoted this fact by writing a = /?, mod a. i) In particular, if a be a number of a, we write a = o, mod a. The appropriateness of these symbolic expressions is made evident by Chap. XI, Th. 7 ; for from it we see that the necessary and sufficient condition for a — fi to be a number of a is that it shall be divisible by a. These expressions are capable of many of the transformations to which ordinary congruences between rational integers can be subjected. The congruence 1) leads to a — /? = o, mod a, 2) and conversely 2) leads to 1). The following deductions will be seen to correspond number for number to those given in the case of rational integers (Chap. Ill, §1). Their proofs are so simple that they will be left to the reader. For them we fall back upon our original definition of a = /?, mod a, as meaning that a — (3 is a member of a, or, what is the same thing, that the principal ideal (a — /3) is divisible by a. Observe the similarity between this and the method employed in the case of rational integers, where we made use of our original defi- nition of a = fr, mod m, 1 Hilbert : Bericht, Cap. III. 323 324 CONGRUENCES WHOSE MODULI ARE IDEALS. as meaning that a — b is divisible by m. i. // ol = /3, mod a, and /? = y> m °d a, then a = y, mod a; for, if a — (3 and /? — y be numbers of a, a — (S -\- fi — y, = a — y, is a number of a. The infinite system of integers of the realm which are con- gruent to a given integer, and hence each to each, mod a, are said to form a number class, mod a. ii. // as=/3, mod a, and yE=8, mod a, then a ± y S3 j3 ± 8, mod a. iii. // ass/}, mod a, then ^a = /x/?, mod a. iv. // a = /?, mod a, and y = 8, mod a, then ay = /?8, mod a; and, in particular, if a = /?, mod a, fftlff a* = /3«, mod a. v. // /(*) = a *» + a^- 1 + • • • + an, &£ a polynomial in x, whose coefficients are any integers of the realm, and if /3==y, mod a, then /(/?) =/(y), mod 0. vi. // fia = (xp, mod a, 3) Men a = /?, mod a/b, where b W f/i<? greatest common divisor of (/*) and a. CONGRUENCES WHOSE MODULI ARE IDEALS. 325 For let (fi) = bm and a = bb, where m and b are prime to each other; then, since fi[a — (3] is a number of a, bm(a — /3) is divis- ible by bb. Hence m(a — £) is divisible by b, and therefore, since m is prime to b, (a — /?) is divisible by b (Chap. XI, Th. 10, Cor.). We have, therefore, since b = a/b. a = /?, mod a/b. In particular, if ji be prime to a, then aE=/?, mod a. Hence in this case the congruence i) may be divided by /x. This indeed is an immediate consequence of the fact that the greatest common divisor of (jx) and a is (i) ; for then there is a number /*£ of (./a) and a number y of a such that /^ + y=i; that is, there exists an integer £ such that /*!e=i, mod a. 4) Multiplying the congruence 3) by £, we obtain a = /J, mod a. Conversely, if there exists a number |, which satisfies the con- gruence 4), the greatest common divisor of (fx) and a is (1) ; that is, (/x) is prime to a. vii. If ct^/3, mod a awe? b be a divisor of a, then a = j3, mod b. viii. If a = /3 with respect to each of the moduli a u a 2 , •••,(!«, */*£» az==P, mod I, zvhere I w ffc /<?a,rt common multiple of a lt o 2 , • ••,(!«. ix. // a = /?, mod a, f/z^n (a) and (/?) &az^ £/^ saw*? greatest common divisor with a; 326 CONGRUENCES WHOSE MODULI ARE IDEALS. that is, all numbers of the same number class, mod a, have the same greatest common divisor with a. Let b be the greatest common divisor of (a) and a. Then, since b is a divisor of a, we have by vii a ===/?, mod b. But a = o, mod b, and hence P = o, mod b. In particular, if any number of a class, mod a, be prime to a, then all numbers of this class are prime to a. § 2. The Norm of an Ideal. Classification of the Numbers of an Ideal with respect to Another Ideal. If we separate the integers of a realm into classes with respect to an ideal, a, of the realm, putting two integers into the same or different classes according as they are congruent or incongruent to each other with respect to a, then the number of these classes is called the norm of a, and is denoted by n[a\. This definition of the norm of an ideal is seen to be in accord- ance with the principal property possessed by the absolute value of the norm of an integer. We shall show later that the original definition of the norm of an integer as the product of an integer by its conjugate has also its analogue in the case of ideals. A system of numbers formed by selecting one from each of the classes formed as above with respect to an ideal, a, is called a complete system of incongruent numbers, mod a, or a complete residue system, mod a. There are evidently in such a system exactly n[a] numbers. Instead of separating all the integers of a realm into classes with regard to their congruence with respect to an ideal, we may consider simply the numbers of a single ideal, a, and put two of these numbers, a lf a 2 , into the same or different classes with respect to an ideal, b, according as we have OLi^a.2, mod b, or a x ^p.a 2 , mod b. We shall denote by the symbol {a, b} the number of such classes into which the numbers of a fall with respect to b. 1 1 See Dirichlet-Dedekind : § 171. CONGRUENCES WHOSE MODULI ARE IDEALS. 327 Evidently {a, b} is not greater than w[b], since a does not com- prise all integers of the realm k unless a=(i), in which case {(i),6} = «[6]. It will be interesting to make use of this classification of the numbers of one ideal with respect to another ideal to prove an important theorem (see p. 336) and we proceed now to prove the following relations : i. {a, *} = {a, a — b}. ii. {a, h} = {a + h, b}. iii. {a( v ), b(^)} = {a, b}. iv. {a, c} = {a, &}{&, c}, where a is a divisor of b, and b a divisor of c. i. To prove {a, &} = {a, a — &}. We observe that a — b, the least common multiple of a and b, is composed of all numbers common to both a and b. Hence, if a lt a 2 be two numbers of a such that 0^ = 32* m °d b, that is, such that a x — a 2 is a number of b, then, since a t — a 2 is also a number of a, it must be a number of a — b, and therefore a x =a 2 , mod a — b. Conversely, if a x = a 2 , mod a — b, then a x — a 2 is a number of b; that is, a x ^a 2 , mod b. Hence any two numbers of a, that are congruent to each other with respect to b, are congruent to each other with respect to a — b and vice versa. Therefore we have {a, h} = {a, a-h}. ii. To prove {a, b} = {a-f b, b}. Let a!,a 2 , ••-,a m (m = {a, b}) 1) 328 CONGRUENCES WHOSE MODULI ARE IDEALS. be a complete system of incongruent numbers of a with respect to b. Then every number of a + 8 is congruent to one of these numbers with respect to ft, for all numbers of a + B can be written in the form a + /3, where a is a number of a and /3 a number of B. And from a^di, mod fc, where a* is one of the numbers 1), we have a-\- /3 = oii, mod B, since /? = o, mod ft. Moreover, since a + ft contains all the numbers of a, some numbers of a + & will be congruent to each one of the integers of the system 1), mod ft. Hence {a, 6} = {a + 6, 6}. iii. To prove {aW,BW} = {aJ}. Let a 1 ,a 2 ,--,a m (m={a, &}) be a complete system of incongruent numbers of a with respect to b, then G^iy, a 2 r), • • • , a m ?/ form a complete system of incongruent numbers of a(>y) with respect to the mod 6(77) ; for they are all incongruent, mod &(*/), to each other, since, if a g r]==a h r), mod b(^), then a g z==ah, mod &, which is impossible. Furthermore, every number of a(rj) is con- gruent to one of these integers, mod b(^), for, if ay be any num- ber of a(rj), and a = at, mod b, then (a — a.i)r) is a number of &(??), and hence arj^ai-rj, mod 16(17). Hence {<*(,), &«} = {a, b}. CONGRUENCES WHOSE MODULI ARE IDEALS. 329 iv. To prove that, if a be a divisor of b and b a divisor of c, then {a, c} = {a, h}{h, c}. Let a 1 ,a 2 ,'~ t a m (tn={a, fy) 2) be a complete system of incongruent numbers of a with respect to the modulus b, and let ft, ft, — ,ft(n = {6, c}) 3) be a complete system of incongruent numbers of b, mod c. We shall show that the mn numbers [ r= 1,2, ••-,*» Zr + Psi 4) L$ = 1,2, •••,M which are all evidently numbers of a, form a complete system of incongruent numbers of a, mod c. The numbers 4) are incongruent each to each, mod c; for, if a a + Pb = a c + p d , mod c, 5) then, since b is a divisor of c, a a -f fii sb a c + /?<?, mod b, and hence, since fib and pa are numbers of b, a a ==a c , mod b, which is impossible unless a a = a c . But, if a a = a c , then from 5) we have p b ==p d , mod c, which is impossible. Hence the numbers 4) are incongruent each to each, mod c. Moreover, every number, a, of a is congruent to some one of the numbers 4), mod c; for suppose a^a-x, mod b, where a t is one of the numbers 2), then a — a» is a number of b, and we have a — a<ss/b, mod c, 330 CONGRUENCES WHOSE MODULI ARE IDEALS. where p h is one of the numbers 3), and hence a = a* + fa, mod c, where a« + fa is one of the numbers 4). The numbers of a complete system of incongruent numbers of a, mod c, are therefore exactly mn in number, and hence {a, c} = {a, &}{*>, c}. Theorem i. If i lt = a 1 o) 1 -\- a 2 w 2 , i 2 , = &!&>! + b 2 <a 2 , £?£ a &a«'j 0/ f&£ wfea/ a, £/z£ absolute value of the determinant of the coeffi- cients a lf a 2 , b lf b 2 is equal to the norm of a; that is, «[a] = Let "1 "2 where a<o lt b(o ± + cw 2 is a canonical basis, a and c being taken positive. Since 1 ^2 it is sufficient to show that a o b c = ac (Chap. XI, § 2), w[q] =ac. In the expression Um x -j- ^a> 2 6) let w run through the values o, 1, •••, a — 1, and v through the values o, 1, •••, c — 1. We shall show that the ac numbers so formed constitute a complete system of incongruent numbers with respect to a. m They are incongruent each to each with respect to a; for, if u 1 <o 1 + v x oi 2 and u 2 & x + v 2 o> 2 be any two of them, and Wjtoi -\- v 1 (o 2 ss WjCDj -f- z> 2 w 2 , mod a, then {u x — u^^-^- (v 1 — v 2 )(o 2 ^o, mod a, and hence, since c is the greatest common divisor of the coeffi- cient of <o 2 in all numbers of a, 1 Hilbert: Bericht, Satz 19. CONGRUENCES WHOSE MODULI ARE IDEALS. 33 1 v x — v 2 = 0, mod c. But v x and v 2 are both less than c, hence v x = v 2 . It follows that (u x — u 2 )a> x ===o, mod a, and hence, since a is the greatest common divisor of the coeffi- cients of o) x in all numbers of a in which the coefficient of <o 2 is o, u x — u 2 ==o, mod a. But u x and u 2 are both less than a, hence u x = u 2 . ThUS U x 0) x -f" ^iW 2 = W^ + Z>2 W 2> and the numbers 6) are incongruent each to each with respect to a. Moreover, every integer of the realm is congruent to one of the numbers 6) with respect to a. For, let (0 = t x oi x -j- t 2 oi 2 be any integer of the realm, and let t 2 = mc + r 2 , where m and r 2 are rational integers and r 2 satisfies the conditions ogr 2 <c. Also let t x — mb = na-\-r x , where n and r x are rational integers and r x satisfies the conditions g r x < a. Then *1 W 1 + ^2 W 2 = {™b + na + r i) W l + ( WC + r 2) W 2 = waw! -j- m(bo) x + ^w 2 ) + r i w i + r 2 w 2> and hence f jo^ -J- t 2 w 2 h r x o) x + r 2 w 2 , mod a. But r^ + r 2 w 2 is one of the numbers 6). Hence every integer of the realm is congruent to one of these 332 CONGRUENCES WHOSE MODULI ARE IDEALS. numbers with respect to a, and therefore, since they are ac in number n[a] = ac. Hence n[a] = K b i\ From this theorem we see that the norm of an ideal is always finite. Ex. Since 7, 3 + y/ — 5 is a basis of the ideal (7, 3 + V — 5)> 7 o »(7i 3 + V— 5! 3 « = 7. In the case of non-principal ideals, we shall omit [ ] and write merely n before the symbol to denote the norm, as in the example just given. Cor. 1. Since, if a 1 w 1 + o 2 o> 2 , b 1 a) 1 + b 2 <o 2 be a basis of a, then a i w i' + a 2 «>2> fri w i' + ^2^2' W a basis of a' (Chap. XI, § j), we have «[v] = a, a n h K n [a] . Cor. 2. If (a) be a principal ideal, where a is a rational in- teger, then n[(a)]=a 2 ; for awj, aa> 2 is a basis of (a), and hence a o »[(*)] = O a = <r. We can prove by this method that the norm of any principal ideal (a) is equal to the absolute value of the norm of the integer a which defines (a) ; that is n[(a)] = \n[a]\. But a simpler proof can be found, based upon a theorem to be given later. Cor. 3. // a,= (a x <a x + a 2 w 2> &i w i + b 2 <o 2 ), be an y ideal and a \ a 2 K K =4a] ' then a 1 <a 1 -j- a 2 w 2 , b x ta r -J- fr 2 w 2 £y a basis of a. CONGRUENCES WHOSE MODULI ARE IDEALS. 333 Theorem 2. If a = bc, zvhere b and c are any ideals, there are exactly n[c] numbers of a complete system of incongruent num- bers, mod a, which are divisible by b. Let rify»""»Y»['i 7) be a complete system of incongruent numbers, mod c, and let /? be a number of b such that (/?)/& is prime to c (Chap. XI, Th. 11). The numbers are incongruent each to each, mod a; for, if py h ==f$yi, mod a, then y ft = y<,mod c (§1, vi), which is impossible. Moreover, every integer, @ lt divisible by b is congruent, mod a, to some integer of the form n/3, for since b is the greatest common divisor of a,= (a lf a 2 ), and (/?), we have K=(a 1 ,a 2 ,p), whence, since ft is a number of b, it follows that where | x , | 2 and fx are integers of the realm, and hence P 1 == f i(3, mod Q. But every integer of the form fif3 is congruent, mod a, to some one of the numbers 8) ; for fi is congruent to some one, say, -/», of the numbers 7), mod c, and from fj.^y if mod c, it follows easily that fifx==l3yi, mod a. Since, also, every integer congruent to one of the numbers 8), mod a, is divisible by b (§1, vii), and the numbers 8) are «[c] in number, there are in every complete system of incongruent num- bers, mod a, exactly n[c], = n[a]/n[b], numbers that are divis- ible by b. 334 CONGRUENCES WHOSE MODULI ARE IDEALS. Theorem 3. The norm of the product of two ideals, a, b, is equal to the product of their norms. 1 Let a be a number of a such that the quotient (a) /a is prime to 6 (Chap. XI, Th. 11). Let CLi,CL 2 > •">an[a] 9) and Pi,P 2 ,-",Pnif>i 10) be complete systems of incongruent numbers with respect to a and b, respectively. Then the w[ct]n[b] numbers of the form where £ and 17 run through the values 9) and 10), respectively, form a complete system of incongruent numbers with respect to ah, and hence are n[ab] in number. To show this it is necessary and sufficient to show first that no two of the integers 11) are congruent to each other with respect to the modulus ab, and second that every integer of the realm is congruent to one of them with respect to ab. Let api -\- ai and a/3j -f- a m be any two of the integers 11). If afii + ai = a/3j + a m , mod ab, 12) then a(pi — pj)-\-a% — a m = o, mod a, and hence, since a ((3i — fij ) = o, mod a, we have ai — a m = o, mod a, whence a.i = a m . Then from 12) it would follow that a (pi — pi) = o, mod ab, and hence, since (a) + ab is a, /3i — pj = o, mod b, which is impossible unless pi = pj. 1 Hilbert: Bericht, Satz 18. CONGRUENCES WHOSE MODULI ARE IDEALS. 335 Therefore 12) is impossible and the integers 11) are incon- gruent each to each, mod ah. Moreover, if to be any integer of the realm, we have a« = w, mod a, 13) where a 8 is one of the integers 9). Now from 13) it follows that w — a 8 is divisible by a. But every integer of a complete residue system, mod ah, that is divis- ible by a is congruent to one of the integers ap lt ap 2i •••,a0„ [6] , 14) mod ah (Th. 2) ; that is, the integers 14) are representatives of all and only those incongruent number classes, mod ah, whose numbers are divisible by a. Hence we have w — a 8 ^a(3 r , mod ah, whence <o = a/? r -f-a g , mod ah, where af3 r + cl 8 is one of the numbers 11). The numbers 11) form therefore a complete system of incon- gruent numbers, mod ah, and hence i n[ah] =n[a]n[h]. A complete system of incongruent numbers, mod ah, fall int 11 [a] classes each containing n[h] numbers, such that the numbers of each class are congruent each to each, mod a, but the numbers of any class are incongruent to all those of any other class, mod a. We may arrange these classes as follows: aft + a tt ap 2 + a lf • • •, a0„ [6] + a lt ft a Pi + 3n[ a ],a/? 2 + a «[a]» •• •><*/?«[*] -f-a» [a] , where a,a lf a 2 , • • -,a» [«],&,&> " m 9pm*i are as defined above. It will be seen that the numbers of each row are all and only those of the complete system of incongruent numbers, mod ah, that are congruent to each other, mod a. 33^ CONGRUENCES WHOSE MODULI ARE IDEALS. There are, therefore, exactly n[b] numbers of a complete residue system, mod ab, that are congruent to any given number, mod a. In particular there are, as we have already seen, exactly n[b] numbers of a complete residue system, mod ab, which are divisible by a. It will be interesting to obtain by means of the development of § 2 another proof of the above important theorem. We begin by proving that {a, ab} =*[&]. ' Let a be a number of such that ah -j- (a) = a; then* ab—(a)=b(a), for the least common multiple of two ideals is equal to their product divided by their greatest common divisor. We have now {(a), ab} = {(a)-f ab, ab} (§2, ii) = {a, ab}, and also {(a), ab} = {(a), (a) — ab} (§2, i) = {(«), (a)b} = {(1), b} (§2, iii) = *[&]. Hence {a, ab}'=n[b]. To prove the theorem, we observe that, since (1) is a divisor of a, and a is a divisor of ab, we have by § 2, iv {(i), ab} = {(i), a}{a, ab} and hence n[ab] =n[a]n[b]. We have seen (Chap. XI, Th. 5) that the product of an ideal, a, by its conjugate, a', is a rational principal ideal (a). We shall now show that n[a] = \a\; or in other words, Theorem 4. // a be an ideal of a quadratic realm and a' its conjugate, then aa' = (n[a]). We have act' = (a) (Chap. XI, Th. 5), where a is a rational integer which may be assumed to be positive. Hence n[a]n[a'] =n[(a)] *=a 2 (Th. i.Cor.2). CONGRUENCES WHOSE MODULI ARE IDEALS. 337 But n[a'] =n[a] (Th. i,Cor. i). Hence n[a]=a, and aa'=0[a]). This theorem for the general realm of the wth degree is that aa'a" ••• a*"- 1 ) = 0[a]), where a', a", ..^at"- 1 ) are the conjugates of a. The proof in the case of the quadratic realm here given is much simplified by having seen (Chap. XI, Th. 5) that in a quadratic realm the multipli- cation of a by a' gives a principal ideal. See Hilbert : Bericht, p. 191. This property of the norm of an ideal might be taken as its definition. It would then be exactly in line with that of the norm of an integer. From Th. 4 it is evident that n[a] is divis- ible by a, as in the case of integers. Theorem 5. The norm of a principal ideal, (a), is equal to the absolute value of the norm of the integer a defining the ideal; that is, »[(a)] = |«[a]|.' Let (a) be any principal ideal and (a') its conjugate. Then (a)(a') = (»[(a)])(Th. 4), and also (a) (a') = (aa f ). But aa' = n[a]=a, a rational integer, since the norm of an algebraic integer is a rational integer, and n[(a)]=b, a positive rational integer. Hence ( a ) = (fc). Since a is therefore divisible by b, and b by a, we have \a\=b, and hence »[(*)] = Ma]|, ' x Hilbert: Bericht, Satz 20. 22 338 CONGRUENCES WHOSE MODULI ARE IDEALS. Theorem 6. The norm of a prime ideal, p, is a power of the rational prime which p divides. 1 Let 1, w be a basis of the realm and p = (a, b -j- c&), where a, b -f- Co) is a canonical basis of p. It is evident that a is a prime, for, if a=a x a 2 , then since p divides a, it must divide either a x or a 2 , say a lt then a t would be a number of p, which would be contrary to the hypothesis that a, b -\- Cw is a canonical basis of p, and hence that a is the smallest rational number of p. Hence a is a prime, p. We have then (P)=pa, whence n [(P)] =w[^)]«[a], and P* = n[p]n[a], (Th. 1, Cor. 2). Hence, since n[£] and n[a] are positive rational integers, we have either n[P]=P, 15) or n [p]=p 2 ; 16) we call p a prime ideal of the first or second degree according as 15) or 16) occurs; that is, the norm of a prime ideal, p, is a power of the rational prime which p divides, and the exponent of this power is called the degree of p. For example: and hence (3, 1 +V — 5) is a prime ideal of the first degree; on the other hand, n[(2)]=2 2 = 4 , and hence (2) is a prime ideal of the second degree, both (3, 1 -f-V — 5) and (2) having been shown to be prime ideals. Cor. 1. In a canonical basis, p, b -f- c<n, of a prime ideal, p, the coefficient c is 1 or p, according as p is of the first or second degree. 1 This theorem holds for realms of any degree, but the method of proof used here is not applicable to those of degree higher than the second. See Hilbert: Bericht, Satz 17. CONGRUENCES WHOSE MODULI ARE IDEALS. 339 Ex. i. If a and b be two ideals and a be prime to «[b], then n[a] is prime to n[b]. Ex. 2. If J>i, p 2 , ••-, pn be prime ideals of the first degree no two of which are conjugate, and whose norms are pi, p 2 , •••, p n , show that the smallest rational integer in the product pip 2 • • • pn is pip 2 • • • pn. Ex. 3. If the ideal a does not contain the factor (p), where p is a rational prime, and n[a] be divisible by p n but not by p n+1 , then a is di- visible by p n , where n[p] = p. § 3. Determination and Classification of the Prime Ideals of a Quadratic Realm. The last theorem furnishes us with a method for obtaining and classifying the prime ideals of any quadratic realm, &(Vra), similar to that employed for the prime numbers of &(/),£( V — 3) and &(V 2 )- We have seen that every prime ideal divides a rational prime; hence, to obtain all prime ideals of k(^/m) we need only factor all rational primes into their prime ideal factors in k(ym). If p be a prime ideal and p the rational prime which p divides 1 (since ( — p) = (p) we may assume p positive), there are, it has been shown, two cases to be distinguished. That is, if then P 2 = n[P]n[\], and we have either i. n[p]=p ; n[\]=p, or ii. n [p]=p 2 ; «[j]=i, and hence j= (1). From i it follows by Th. 4 and the unique factorization theorem that (p)=pp f ; that is, j = to'; and from ii that (P)=9- 1 That only one rational prime can be divisible by a prime ideal p is evident from the fact that, if two primes p and q were divisible by p, then their rational greatest common divisor 1 would be a number of p, and p would be (1). 340 CONGRUENCES WHOSE MODULI ARE IDEALS. In i, (/>) is factorable into two conjugate prime ideals of the first degree. In ii, (/>) is a prime ideal of the second degree. We shall now determine the relation which the form of p bears to the occurrence of these cases, and shall see that the factorization of (p) depends upon whether the discriminant of the realm is a quadratic residue, a quadratic non-residue, or a multiple of p. We shall show first that the necessary and sufficient condition for the factorability of (p) is that d shall be a quadratic residue of p or divisible by p, hence proving incidentally that the condi- tion for the non- factorability of (p) is that d shall be a quadratic non-residue of p. Suppose that i occurs; that is, {p)=W- i) Since n[p] ==p, there are p incongruent number classes with respect to p. We may take as representatives of these classes the numbers o, I, •••, p — i; for, since p is the smallest rational number in p, the differences of no two of these numbers is a number of p s and they are therefore incongruent to each other with respect to p. It is evident that y/m, which is an integer, is congruent to one of these numbers, say a, with respect to J>; that is, a — \/m = o, mod p, therefore, since a+\/ra is an integer of k(^tn), (a — V m ) («+V w ) =a 2 — m==o, mod p, and hence, since a 2 — m is a rational number and p the smallest rational number in p, a 2 — m = o, mod p. 2) Hence that m shall be a quadratic residue of p or divisible by p is a necessary condition for the factorability of (/>). We must now distinguish between the two cases p=\=2 and p==2. CONGRUENCES WHOSE MODULI ARE IDEALS. 34 1 First let p =j= 2. It may be shown that in this case a 2 — m zaa o, mod p, is a sufficient as well as necessary condition for the factorability of (p) ; for from a 2 — m=(a— yra)(a+yra)=o, mo d p, it follows (Chap. XI, Th. 9) that, if (p) be unfactorable, either a — ym = o, mod (p), or fl-f\/w = o, mod (/>), and hence either 7 — .r-f 1/1///Z # — ym = - — p ,— x + y\/m or « -f 1/ w = - p 3) where .ar and y are either both even or both odd, the latter case being possible only when fgeel, mod 4. The equations 3) lead to the impossible equations Hence 3) are impossible, and that m shall be a quadratic residue of p or divisible by p is a sufficient as well as necessary condition for the factorability of (p). Therefore that m shall be a quad- ratic non-residue of p is a necessary and sufficient condition for the non- factorability of (p). Now let the symbol (n/q), where q is an odd rational prime and n any rational integer, denote 1, — 1, or o, according as n is a quadratic residue or non-residue of q, or a multiple of q. We shall now obtain the factors of (p) when (p)=pp', and shall show that when (m/p) = 1 they are different, and when (m/p)=o they are alike; that is, (p) is then the square of a prime ideal. 342 CONGRUENCES WHOSE MODULI ARE IDEALS. When (m/p) = i, a is not divisible by p, and we shall show by actual multiplication that (P) = (P, a +V») (p, a—y/m). We have (Pi a-\-\/^n)(p, a — V m )=(P 2 > P a — P^/m, pa + PV m > a2 — m ) =(p 2 , pa — py/m, 2pa, a 2 — m) =(p 2 , pa — p\/m, 2pa, a 2 — m, p) since p is the greatest common divisor of p 2 and 2pa and may therefore be introduced into the symbol. We shall show now that (P, o+V»i) + tt a—y/m). If they were the same, both would equal (Pi a-\-\/m, a — -\/m) = (p, a-\-y/m, 2a) (f=(fc a+y/ni, 2a, i) = (0, since p and 2a are two rational numbers prime to each other and i may therefore be introduced into the symbol. Hence (p) is the product of two different conjugate prime ideals when m is a quadratic residue of p. When (m/p) =o, a is divisible by p, and we have by similar analysis (P) = (P, V») (Ps — Vw) = (P, Vm) 2 . Hence (p) is the square of a prime ideal, when m is divis- ible by p. We see that, since the discriminant of the realm, d, = m or \m, according as m== I, mod 4, or = 2 or 3, mod 4, (d/p) = (m/p). We may express the results so far obtained conveniently as follows : CONGRUENCES WHOSE MODULI ARE IDEALS. 343 // p be an odd rational prime, (/>) is the product of two differ- ent conjugate prime ideals, or is itself a prime ideal, or is the square of a prime ideal, according as (d/p) =1, — i, or o. To obtain basis representations of p we make use of Th. I, Cor. 3, and at once recognize that when (m/p) = i and m = 2 or 3, mod 4, (P, «+Vw) is the required representation, for p o a I In the case ra=i, mod 4, (p, o+V w ) is n °t a basis repre- sentation of p, for when we express a -\- yjm as a linear combi- nation of the basis numbers 1,(1 -\-'\Jm)/2 of the realm, we have that is not a basis representation, since / o" a— 1 2 = 2/**|>]. In this case we can, however, get a basis representation of p as follows: since p is odd, a can be chosen so as to be not only a root of a 2 = m, mod p, but also odd. Supposing this done, we can introduce into the symbol of p the number (a-\-^/m)/2, and then omit a -f-\/m, obtaining a + Vm ( a+Vm\ = (A- — j (a — 1 1 + |/«* \ which is a basis representation of p, since P ° #— 1 2 = / = »[>]. 344 CONGRUENCES WHOSE MODULI ARE IDEALS. We consider now the case (tn/p) = o. In the cases m = 2 or 3, mod 4, we have as the required basis representation p=(p, V«) since o 1 -/-*[»] When m=i, mod 4, we can introduce the number (p +y*»)/2 into the symbol (p % ,y/m), since p is odd, and thus have — / — p + -j/w \ / p- 1 1 + i/;;A as a basis representation, since 1/ o = / = *!>]■ Let now p = 2. We have in all cases (w/2) = 1 or o; that is, the necessary condition for the factorability of (2) is always satisfied. As to the sufficiency of this condition we must however distinguish three cases according as m = 3, 2 or 1, mod 4. When ms=3, mod 4, we have (in/2) = 1, and from 2), 0=1. Putting, therefore, in equations 3) p = 2 and a=i, and re- membering that when m = 3, mod 4, # and y must both be even, we see that 3) leads to the impossible equation ± I = 2;r. Hence (m/2) = 1, in the case ^ = 3, mod 4, is a sufficient condi- tion for the factorability of (2). We have indeed (2) = (2, 1 +Vw) (2, 1 — y») for 2 and 1 +V W are evidently numbers of J) and 2 o CONGRUENCES WHOSE MODULI ARE IDEALS. 345 Hence (2, 1 + ym) and (2,1 — y/ni) are the factors of (2). But evidently (2, i+V"w) = (2, 1— Vw), and hence (2) = (2, 1+Vw) 2 , a result which may be verified by multiplication. Thus when m==3, mod 4, (2) is the square of a prime ideal. When m = 2, mod 4, we have (m/2) =0, and from 2) a = o. Putting, therefore, in 3) p = 2 and a = o, and remembering that when m = 2, mod 4, .r and y must be even, we see that 3) leads to the impossible equations ± i=2y. Hence (m/2)=o is also a sufficient condition for the facto ra- bility -of (2). We can show just as above that in this case (2) = (2, V"0 2 . When mm, mod 4, we have (m/2) = 1, and from 2) o=I. Putting £ = 2 and a=i in 3) we see, however, that x=i, y = — 1 satisfy the first of these equations and x=i, y = 1 the second, (1 — s/m)/2 and (i-j-\/w)/2 both being integers of k(^m), when m=i, mod 4. Hence both (1 — ^m) and (i-|-yra) are divisible by (2) and nothing is known as to whether (2) is prime or not. To determine when (2)=pp' we may proceed as follows: If (2) = pp', then o, 1 is a complete system of incongruent num- bers with respect to p, and hence (1 -\-\/m)/2 must be con- gruent to either o or 1 with respect to p; that is, we must have either 1 + ^™ A c a o, mod £, 1 -I- i/»i 1 — Vm or 1 = = o, mod to ; 2 2 and hence in any case 34^ CONGRUENCES WHOSE MODULI ARE IDEALS. But (1 — m)/4 is a rational integer and we must have therefore 1 — m = o, mod 2, 4) since 2 is the smallest rational number in p. From 4) it follows that 1 — m = o, mod 8 ; that is, rn=i, mod 8, is a necessary condition for the factorability of (2) when w=i, mod 4. We must now distinguish two cases according as m=i or 5, mod 8. In the latter case (2) is evidently a prime ideal, for 4) is no longer satisfied. We shall proceed to show that when ra=i, mod 8, (2) is the product of two different conjugate prime ideals. If (2) be factorable, p must contain one of the numbers (1 -\-\Zm)/2, (1 — \/m)/2, and hence p' the other. Moreover, we have || 2 Oil Ilo l \\ Hence, if (2) be factorable, we have / 1 -f Vm\ ( 1 — \/m\ (2) = ^, -__j|v__ r _j, and this may be shown to be correct, for by multiplication we get / 1 + Vm\ ( 1 — Vm\ ( .— .— 1 — m\ \2 % -j— J (s, - -^— j = (^4, 1 - Vm t 1 + •*, — - J = U, 1 - i/«, - -, 2 J _=(2), since (1 — m)/4 and 1 — s/m are divisible by 2, when m=i, mod 8. Moreover, / 1 + Vm\ , / I — l/w\ CONGRUENCES WHOSE MODULI ARE IDEALS. 347 for, if they were the same, they would both equal (1 + Vm 1 — \/m \ ( 1 4- v m I — Vm, \ which is, of course, impossible. Hence, when m= i, mod 8, (2) is the product of two different conjugate prime ideals. We may collect the results obtained for (2) as follows: (2) is the square of a prime ideal when m = 3 or 2, mod 4; it is the product of two different conjugate prime ideals, when m == 1, mod 8, and it is a prime ideal when m = 5, mod 8. We have evidently as basis representations of the factors of (2) in these cases respectively (2) = (2, 1 +\/m) 2 , (2) = (2, ym)\ I + Vm \ ( 1 — Vm , . / 1 + Vm\ ( 1 — Vm\ (2) = (2, 1 -fV w )- Let now the symbol (w/2) denote 1, — 1, or o according as n is a quadratic residue or non-residue of 8 or is divisible by 2, and observe that, when m = 3 or 2, mod 4, d = 4m, and hence is always divisible by 2, and that when m=i, mod 4, d = m, and hence is a quadratic residue of 8 when and only when w=i, mod 8, and a quadratic non-residue of 8 when and only when m=5, mod 8. We may now combine the results obtained for p = 2 with those for p =4= 2 in the following theorem : Theorem 7. // p be any rational prime, (p) is the product of two different conjugate prime ideals of the first degree, a prime ideal of the second degree, or the square of a prime ideal of the first degree, according as (d/p) = J, — /, or o. 1 An ideal a of a quadratic realm such that a = a' and which con- tains as a factor no ideal (a), zvhere a is a rational integer differ- ent from ± 1, is called an ambiguous ideal. The ambiguous prime ideals of a quadratic realm are evidently the prime factors of (d). The following table gives basis representations of the prime factors of (p) in a convenient form for reference. x See Hilbert: Bericht, Satz 97. 348 CONGRUENCES WHOSE MODULI ARE IDEALS. In it a satisfies the congruence a 2 = m, mod p, and is, more- over an odd integer in the case when w= i, mod 4. ©-■ (,')- m = I, mod 4 m hi 2 or 3, mod 4 Ex. 1. *(V-i3) We have — 13 =3, mod 4, whence 1, V — 13 is a basis of &(V — 13) and d = — 52. Since and i 2 as — 13, mod 2, we have (2) = (2, 1 + V — 13) 2 . Since (3) is a prime ideal. Since (5) is a prime ideal. Ex. 2. Find basis representations of the prime ideal factors of all rational primes less than 20 in the realms fc(V — 7), fc(Vn) and *(V30~). Ex. 3. If the norm of any ideal be divisible by an odd power of a rational prime, p, then p is factorable into two conjugate prime ideals of the first degree. § 4. Resolution of any Given Ideal into its Prime Factors. We have in the last section given a general method for resolv- ing any principal ideal defined by a rational prime number into its prime ideal factors. The resolution of any given ideal a can be effected by observ- ing that the product of the norms of the prime factors of a must equal n[a], and hence the only possible prime factors of a are the prime ideal factors of the rational primes which divide n[a]. CONGRUENCES WHOSE MODULI ARE IDEALS. 349 We then determine by actual multiplication which of the finite num- ber of prime ideals satisfying this condition are the proper ones. We shall see that the resolution of any ideal a,= (a lf a 2 , • • • , a„) , can be made to depend upon the resolution of the principal ideals (aj, (a 2 ), •••, (a„), and shall illustrate by the following ex- ample the resolution of a principal ideal into its prime factors. Let &(V — 5) be the given realm and (io+V — 5) be the given ideal; then n[(io+V zr 5)] = io5 = 3-5-7- Hence (io-f-V — 5) must be the product of three prime ideals whose norms are respectively 3, 5 and 7. The prime ideals whose norms are 3 are evidently (3, 1 -f~V — 5) an d (3, 1 — V — 5)- The only one whose norm is 5 is (V — 5)- Those whose norms are 7 are (7, 3 +V ZI 5) and (7, 3— V^S). By multiplication we can determine which of the four possible combinations of these ideals is the correct one. We can, however, materially shorten the process by observing that, if (10+V — 5) be divisible by (7, 3 — V — 5)> tnen ( I0 +V — 5) * s a number of (7, 3— V :zr 5); that is, (7, 3— V=5j=(7, 3— V^5, io + V^) = (7, 3— V=5, io+V=5, 13) = (7, 3— V^ io+V^, 13, 1) = (1), which is impossible. Hence (7, 3 — V — 5) is not a factor of (10+V — 5)- Since 7, 3 — V — 5 is a Das i s °* (7> 3 — V — 5) we could have determined whether or not 10 +V — 5 is a number of (7, 3 — V — 5) by seeing whether or not io+V^5 = 7-r+ (3— V z:r 5)y where x and y are rational integers. This equation gives .a- =13/7, yz= — 1, and it is again proved that (10 +V — 5) is not divisible by (7, 3 — V — 5)- I n like manner we can show that (3, 1 — V — 5) does not divide (10+V — 5)- Hence (io+V=5) = (3> i+V=5)(V=5)(7, 3+V-"5). 350 CONGRUENCES WHOSE MODULI ARE IDEALS. Had we first tested either (7, 3+V — 5) or (3> I +V — 5) we should have found, of course, that (10 +V — 5) was divis- ible by it. If n[(a)] be divisible by a higher power, p r , than the first of a rational prime, p, then either (p) is a prime ideal in which case a is divisible by p r / 2 , this case being possible therefore only when r is even, or (p) is the product of two conjugate prime ideals, p, p', of the first degree. In this case (a) may be divisible by both p and p', and hence a by p, or (a) may be divisible simply by a power of one of the ideals, say p. If a = p e a x , where a x is not divisible by p, then (a x ) cannot be divisible by the product pp' and hence, if w[(«i)] be divisible by p 8 , then a x is divisible by either p s or p' 8 , these cases occurring respectively as (a x ) is divisible by p or p'. The resolution of any principal ideal into its prime factors can therefore be effected. Let now ct= (a lf a 2 , • • •, a n ) be any ideal. Since a is the great- est common divisor of the principal ideals (fltj.), (a 2 ), "*> ( a »)> we can effect the resolution of a into its prime ideal factors by resolving the ideals (aj, (a 2 ), •••, (a„) into their prime factors and taking their greatest common divisor; this will be a. Ex. 1. Let (21, 10 + V — 5) be the given ideal. We have found above that Oo + v=5) = (3, i + V^KV^K;, 3 + V=5), and we have evidently (21) = (3, i + v T -5)(3, i— yf^s)(a t a+V^X?. 3— '•^s). Hence (ai, 10 + V^ =: 5) = (3, i + V"^ r 5)(7, 3 + V" ::r 5) is the resolution of (21, 10 + V — 5) into its prime factors. Ex. 2. Resolve the ideal (30) into its prime ideal factors in the realms KV^S), Hy/^7) and *(V30)._ Ex. 3. Resolve the ideal (24 — V 2 ^) into its prime ideal factors in the realm k{\/26). Results should be verified by multiplication. CONGRUENCES WHOSE MODULI ARE IDEALS. 351 There are many devices which shorten numerical work with ideals, some of which will be illustrated later in the solution of examples. § 5. Determination of the Norm of any Given Ideal. If an ideal has been resolved into its prime factors, or if we have a basis of the ideal, its norm is easily found. Let a,= (a lt a 2 , • • -,a„), be the given ideal, and let be the resolution of a into its prime factors ; then w[a]=«[PiMl>2] --n[pr]. If we have a basis a^ + a 2 w 2 , b 1 oi l -\- b 2 w 2 of 0, we have, of course, at once *[tt] = Theorem 8. The greatest common divisor of the norms of the numbers of a is n[a]. Let n[a] —a, and let a be a number of a such that {a) /a is prime to (a). Then, if a' be the conjugate of a and a' the con- jugate of q, we have (a?) /a' also prime to (a), and hence (n[a])/(a) prime to (a). Therefore a is the greatest common divisor of n[a] and n[a], and hence of the norms of all num- bers of a. 1 It should be observed that the greatest common divisor of the norms of the numbers defining a is not necessarily n[a], though, of course, n[a] is a divisor of it; for example, (1 + V~5, 1 - V 31 !)) = (2, 1 + >/— 5) is an ideal whose norm is 2, but the greatest common divisor of «[i + V — 5] and «[i — V — 5] is 6. § 6. Determination of a Basis of any Given Ideal. Let a, = (a lt a 2 , ••-,«„), be the given ideal and let 11 [a] be known. If two numbers, ai, = a 1 oi x + a 2 o> 2 , dj, = b 1 (D 1 + & 2 w 2 , of a be known, such that 1 2 Hilbert: Bericht, Satz 21. 352 CONGRUENCES WHOSE MODULI ARE IDEALS. then evidently a i} CLj constitute a basis of a. If no numbers sat- isfying this condition be known, we can determine a canonical basis, a, b + Cm, of a, where a and c may be assumed positive, as follows : We observe first that, if a x , b x + c x a> be a canonical basis of an ideal a, and e a rational integer, then a x e, b x e + c x e<a is a canonical basis of the ideal a(e). The determination of a basis of a can therefore be reduced always to the determination of a canonical basis of an ideal which is the product only of prime ideals of the first degree, no two of which are conjugates. Having resolved a into its prime factors, we collect all pairs of conjugate prime ideals of the first degree and all prime ideals of the second degree. The product of these factors will be the principal ideal (e) where e is a rational integer, and we have a = a x (e), where a x is the product of prime ideals of the first degree only, no two of which are conjugates, and whose norms are Pi,P 2 , '",Pm. To find a canonical basis a lf b x -f- c x w of a x , we observe that a x , being the smallest rational integer divisible by a x , must be P1P2 ' • • Pm, and furthermore that, since a x c x = n[a x ]= p x p 2 • • • p m , c x = i. Hence p x p 2 • • • p m , b x + <*> is a canonical basis of a x , where b x is to be determined. Since n[b x -\-ai] is a rational integer and a number of a x we have n[b x + o)] =0, mod p x p 2 ••• p m ; 1) that is, when o)=\/m, b x 2 — m==o, mod p x p 2 . . . p m , 2) and when 1 4- Vm (2d, + i) 2 — m »«-— , 4 -«P, modAA'--A l - 3) It will be easily seen that 2) and 3) have solutions which fall CONGRUENCES WHOSE MODULI ARE IDEALS. 353 into pairs, b 19 — b 1 and 2b 1 + i, — 2b x — i, and that each pair of solutions of 2) gives the numbers b x + y/m, —b x + y/in, and each pair of solutions of 3) the numbers 2b x -f 1 4- Vtn — 2b x — 1 + V'm One of the numbers so obtained must belong to a ± and can, of course, always be determined by resolving the numbers into their prime factors and thus finding out which is divisible by a t . It can, however, usually be determined with much less work from the fact that in determining which of these numbers is divisible by a lt it is helpful to observe that, if a t be divisible by p r but not by p r+1 , where n[p] = p, and if a be one of the numbers satisfying 1), and n[a] be divisible by p r but not by p r+1 , a itself not being divisible by p, then if a be divisible by p, it is divisible by p r . The above method for determining a basis of an ideal a de- pended upon the knowledge of the prime factors of a. We shall now explain how a basis may be determined without this knowl- edge and without that of n[a], giving therefore incidentally a method for finding n[a]. We have seen that, if among the prime factors of a there occur one or more pairs of conjugate ideals, is divisible by a principal ideal (e), where e is a rational integer. Every number, a< + bid), is therefore a number of (e) and hence is divisible by e. Therefore a t and fa must be divisible by e. Conversely, if in every number, ai + bn»j of a.ai and b\ be divis- ible by e, then a is divisible by (e). Let e be the greatest common divisor of the coefficients, a», bi, in all the numbers defining a, and let ai = er iy bi — esi. Then where a x is the product of prime ideals of the first degree, no two of which are conjugates. We have seen that a canonical basis of a x has the form a, b + w. Furthermore a 1 = {r x + ^w, • • • , r„ + s n «>) and the greatest common divisor of r ls •••,r„,^ 1 , •••,£„ is 1. By 23 354 CONGRUENCES WHOSE MODULI ARE IDEALS. multiplying each number, r< + Siio, defining a x , by a>, when a s=sy/m, and by <o — i, when w = -J(i -\-y/m), we can intro- duce into the symbol the numbers, ti + r^ ; that is, such that the coefficient of w is r*. Since the greatest common divisor of the coefficients, r v --,r n ,s x , '-,s n , of w is I, we can find rational in- tegers, Mj, • • •, u n , v lf • • •, v n , such that r x u x -) y r n u n + S& H Y S nVn = I, and hence can introduce into the symbol a number b + w ; that is, one in which the coefficient of to is I. This is evidently one of the desired basis numbers. To find the other number, a, we pro- ceed as follows. Every number in the symbol can be expressed as a linear combination of b-\-o> and a rational integer; thus r 1 + s 1 u = s 1 (b + o>)-Yr 1 — s 1 b = s 1 (b + <o) + c 1 , where c x is a rational integer. We have also c 1 = r 1 + s x < 1 > — s x (b + g>). Hence we can introduce c x into the symbol and omit r x + s x w. Proceeding in this manner with each of the remaining numbers, we have finally in the symbol only rational integers and b + o>. Let a be the greatest common divisor of these rational integers and n[b~Yo)]. Evidently we can introduce a into the symbol and omit all of the rational numbers ; that is, we have a x =(a, & + o>). To show that a, b + w is a basis of a x , we must show that any linear combination a(e x -}- f x a>) -j- (b ~Y <o) (e 2 ~Y / 2 <u) of a and b + o>, where ^ + A^, <? 2 + /2 W are anv integers of the realm, is expressible as a linear combination ax ~Y (b-\-ta)y, where x and y are rational integers ; that is, we must show that the equation ax-Y (b + <»)y=^a(e x -Yf x w) + (b + <o) (e 2 + /*») is satisfied by integral values of .ar and y for all integral values °f ^u /u ^2> A- Multiplying, putting w 2 = m, or <o + J(w — i), according as w=\/ra, or i( l ~YV m )> equating coefficients and making use of the fact that w[fr + w] is divisible by a, we see CONGRUENCES WHOSE MODULI ARE IDEALS. 355 easily that this condition is satisfied. Hence a, b + w is a canon- ical basis of q x . It is well to observe that, when an ideal has the form {a, fc + w), it does not follow necessarily that a, b -f- « is a basis. The necessary and suffi- cient condition for this is that n [b -f- «] shall be divisible by a. Ex. i. Let a=(2)(ii)( 3 , i + V~5> 2 (7, 3+ y— 5) be the ideal whose basis it is required to determine. We have and n[(Xi] = 63. Hence 63, b + V — 5 is a canonical basis of d, where b is to be deter- mined by the condition b + V — 5 — o, mod fe. The condition n[& -f V — 5] ■ o, mod 63 ; that is, b 2 + 5 = o, mod 63, gives fc = 11, — 11, 25 or —25, and hence as possible basis numbers of tti n + \/^5, — H-fV^S. 25 -fV"^, — 25 + V^5- It is easily seen that 11 + V — 5 and — 25 -f V — 5 are not divisible by (3, 1 + V — 5) and hence, of course, are not divisible by cti, while of the two num bers — 11-fV — 5 and 25 -f V — 5 remaining, only — 11 + V —5 is divisible by (7, 3 + y/~^T$). Hence — 11 + V — 5 is the number required, a result easily verified when we see that (- 11 + V~5) = (2, 1 + V^5) (3, 1 + V~5) 2 (7, 3 + V^S). Hence, 63, — 11 + V — 5 is a basis of a„ and (1386, — 242 + 22V — 5) is a basis representation of a. Ex. 2. Let a =(210, 70 + 70V ^5, 90 + 30\/— : 5, — 20 + 40\/^5) be the ideal whose basis it is required to determine. Using the second method, we have e = 10 and a, = (21, 7 + 7>/^ =r 5, 9 + 3V^ r 5 > — 2 + 4V^ r 5)- We see that we can introduce the number 10 -j- V — 5 and have easily a t — (21, 63, 21, 42, 10+ V^S)- Now 21 is the greatest common divisor of 21, 63, 42 and m[io-{- -\J — 5], = 105, and therefore a,= (21, 10+ y/^S), where 21, 10 +V — 5 is a canonical basis. A canonical basis of a is evidently 210, 100 + 10 V — 5. 356 CONGRUENCES WHOSE MODULI ARE IDEALS. § 7. Determination of a number a of any ideal a such that (a) /a is prime to any given ideal m. We have proved the existence of such a number and shall now show how it may be determined in any given case, this problem being not only of interest but of considerable importance in the solution of certain problems to be given later. The proof given above of the existence of a furnishes us with a clue to a method for its determination, which we shall illustrate by some examples. As is seen from the above proof, the determination of a in the general case is dependent only upon its determination in the case where m is a prime ideal p. If a, = (a 1} a 2 , •••,a m ), be any ideal, then some one, a if of the numbers a lf a 2 , • -,a m , x defining a, which are, of course, all divis- ible by a, must be indivisible by Qp ; for otherwise, all numbers of a would belong to ap and a be divisible by ap, which is impossible. This number, a«, is the required number a. We have, therefore, merely to resolve in turn the numbers defining a into their prime ideal factors until we find one which satisfies the required con- dition. Consider the realm &(V — 5) and let a=(2i, io+V=5); m=(2, i+V^). Resolving a into its prime factors, we have a=(3» i+V^X^ 3+V =r 5). Proceeding now to resolve in turn the numbers defining a into their prime ideal factors, we have evidently (2i) = (3)(7) = (3, i+V^5)(3, i-V= :: 5)(7, 3+V= r 5)(7, 3— V=S). We see now that the quotient (2i)/a=(3, 1— V = ^)(7, 3/#V :=r 5), is prime to (2, 1 +V — 5)> an d hence 21 is the number, a, required. 1 We can reduce these always to two but have chosen the more general case so as to show that this reduction is unnecessary. CONGRUENCES WHOSE MODULI ARE IDEALS. 357 Also, since (io + V=7) = (3, i+V=5)(7, S+V^IHV^). The quotient (2i)/a=(3, i— V^)^, 3— V^S) is seen to be prime to (2, 1 +V — 5) 5 hence 10 -f-V — 5 will also serve as a. We could have seen at once that either 21 or 10 -j-V — 5 would serve as the required number, for they are both prime to (2, 1 +V — 5), their norms being prime to n{2, i + V — 5)- If be a principal ideal (/?) and m any ideal, it is evident that the quotient (/8)/(;8) = (i) is prime to m, and hence (3 is the number, a, required. To illustrate the determination of a in the general case, let a=(2i, io+V^) and m=(i5, 5+V^5). Resolving these ideals into their prime ideal factors, we have as above a=(3, i+V— 5) (7, S+V^), and m=(3, 1 — V— : 5) ( V— 7)* the last result being easily obtained by the method employed in the factorization of a, or by simply observing that each number defining m is divisible by V — 5- We have found (21) = (3, i+v=5)( 3 , i-\/-5)(7, 3+V=~5) (7, 3— V— 5). and (lO+V=T5)==(3, i+V=10(7, 3+V— DCV^), and it is well to see whether one of these numbers does not fulfil ' the conditions demanded of a, this often being the case. Here we see, however, that neither of the quotients, (2i)/a=(3, 1— V^)(7 } 3— V^), or (io+V ::=: 5")/a=(V- = T) J 358 CONGRUENCES WHOSE MODULI ARE IDEALS. is prime to m, and therefore that neither of the numbers 21 or 10 -fV — 5 will serve as a. Hence we must proceed to construct a as in the above proof. We have ^ = (3. i+V :zr 5)(7, z+y zzr i)(z, 1— V^), a 2 =(3> l+V=5)<7. 3+V zr 5)(V^5), and it is at once evident that 21 and 10 +V — 5 will serve as a x and a 2 respectively; for the quotient, (2i)/a x =(7, 3—V—5) is prime to ( V — 5)> and (io+y=~5)/a 2 =(i) is prime to (3, 1— V^lO- Hence a = 2i + 10 +y :3 5 = 3 I +V^5 is the number required. This result is easily substantiated by factoring (31 +V — 5) into its prime ideal factors. We have *(3 J +V zr 5) =966 = 2.3.7.23; hence (31 +V — 5) is the product of four ideals whose norms are respectively, 2, 3, 7 and 23. The quotient, (31 +V — 5) At, 1S therefore the product of two ideals whose norms are respectively 2 and 23, and hence is prime to m, whose factors have the norms 3 and 5. We indeed see easily that (3i + V=5) = (2, 1 + V=S) (3. 1 +V : =5) (7. 3 + V=5) (23, 8+V=5). § 8. The ^-Function for Ideals. By <£(m), where m is an)' ideal, we denote the number of integers of a complete residue system, mod m, which are prime to m; that is, the number of integers in a reduced residue system, mod m. CONGRUENCES WHOSE MODULI ARE IDEALS. 359 Thus, if m=(3, i+y^5), taking as a complete residue system, mod (3, 1 +y— 5), the numbers 1, 2, 3, we see that 1 and 2 only are prime to (3, 1 + V^), 3 being divisible by it, and hence <K 3 , i+V ir 5)=2; that is, *(3> i+V :=r 5)=w(3 J i+V^I) — 1. Likewise, if m= (3) = (3, 1 — V^S) (3, I +V :=: 5), taking as a complete residue system, mod (3), the numbers o, 1, 2, V— -5, 1 +V~^ r 5, 2 + V^J, 2 V=5, 1 + 2V zr i, 2 + 2V zr I, we see that 1, 2, V — 5, 2\/ — 5 are prime to (3) and hence *( 3 )=4. - rO-i)O-j0 In particular, we have <f>(i) = 1. Ex. 1. Determine 0(1 -f- V — 5). Ex. 2. Determine 0(13, 5 -f- V — J 4)- Theorem 9. // p be any prime ideal, «ew— ot(i-sj5j). By Th. 2 there are in a complete system of incongruent num- bers, mod p e , exactly M[p e ]/«[p] that are divisible by p, and hence n[p e ] — w[p e ]/«[p] that are prime to p e . Hence «*—[«(« -«&) Ex. We have *(3, I — *'=li)» = «[(3. i-V / =M)3j/i_ ' J ) V »(3i i — V— 14)/ = 2 7 (i-i) =^ / C? The general expression for <£(m), where m is any ideal, could be deduced by a method very similar to the one first employed in R. We shall make use, however, of the second method employed in R (Chap. Ill, § 14), for this was at once applicable in k(i) (Chap. V, § 12), and we shall find the same to be true in the case of ideals. This method depends in R, it will be remembered, 360 CONGRUENCES WHOSE MODULI ARE IDEALS. upon the property of the ^-function that, if a be prime to b, then cf > (ab)=cf>(a).(f)(b). To prove this for ideals we begin by proving the following theorem. Theorem 10. // m be the product of the ideals a lf a 2 , •••,(*« that are prime each to each, and a ly a 2 , --,a 8 any integers of the realm, there exist integers, w, such that wesson, mod a u w = a 2 , mod a 2 , •••, w = a s , mod a s , 1) and all these integers are congruent each to each, mod m. 1 This theorem is proved most easily by a method analogous to the symmetrical one employed for the corresponding theorems in R and k(i). Let m = ct^ = ct 2 b 2 = • • • = a*b«. Then &, + £>,+ ... +b s =(i), and hence there exist in the ideals h lf B 2 , ••-,&« respectively, numbers p x , p 2 , • • • , p 8 , such that Pi+ AH VPs=i (Chap. XI, Th. 8, Cor.). 2) The number a A + a 2 P 2 H h a»P* satisfies all of the congruences 1). For example, we have a A + a 2 p 2 H h 0L 8 Ps ■■ a lt mod a ± ; for, since b 2 , B 3 , ••-,B S are all divisible by a lf the numbers P 2 , p 3 , •-,p s are all divisible by a lf and from 2) it follows that P ± = i, mod a x . Furthermore, if w be any number satisfying the congruences 1), we have by multiplying them respectively with p 1} p 2 , • • •, p s , w^i^aj/?!, mod m, w/3 2 = a 2 p 2 , mod m, 3) o)p8^a s p 8 , mod m. 1 See Chap. Ill, § 14, and Chap. V, § 12 ; also Dirichlet-Dedekind : § 180, II. CONGRUENCES WHOSE MODULI ARE IDEALS. 36 1 Adding together the congruences 3), and making use of 2), we have •1 = aifi t + a 2 p 2 H \- a 8 ps, mod m. Hence all numbers satisfying the congruences 1 ) form a single number class, mod tn. If we let a 1 ,a 2 ,—-,as run through complete residue systems with respect to the moduli a ly a 2 , • • • , a 8 respectively, the resulting ^l>i]tt[a 2 ] -••n[a 8 ]=n[m] values of to evidently form a complete residue system, mod m. The necessary and sufficient condition for w to be prime to m is that a lf a 2 , • • •, a 8 be prime respectively to the moduli a lt a 2 , • • •, a« ; for, from the congruences 1) we see that the necessary and suffi- cient condition that w be prime to each one of the factors a lf a 2 , • • •, q s of m is that each a be prime to its a. Hence, when a lt a 2 , -",a 8 run through reduced residue systems, moduli di, a 2 , ---jCts, respectively, the resulting values of w form a reduced residue system, mod m. We have, therefore, at once the following theorem: Theorem ii. // a lf o 2 , ••-, a 8 be ideals prime each to each, then ^(^q., ••• a.) =<f>(a 1 )<f>(a 2 ) ---<j>(a 8 ). We can now obtain easily an expression for <£(m) when m is any ideal whatever. Theorem 12. // m, = p 1 ei p 2 e2 • • • p/ r , be any ideal, where p 19 p 2 , • • •, p r are the different prime factors of m, then ♦W— M(t- 5 g a )(i.-^ a ) — (»-5Kj)' By Th. 11 we have <f>(m)=<f>(p^)<f>(p 2 ^) •••<A(Pr e 0, from which by Th. 9 it follows that ^)=4^<i-^ ] )«[V]( I -4 ] )--[^]( I -4b) 362 CONGRUENCES WHOSE MODULI ARE IDEALS. Hence by Th. 3 W = 4« 1 ](i-^ j )( I -4j)-( I -^ j ) Ex. 1. We have (21, 10 + V^S) = (3, 1 + V^5) (7, 3 + V^5) and hence 0(21, 10 + V — S) =21(1 — |) (1 — }) = 14- Ex. 2. Find *(6-hV — 14) and 0(189, 77 + 7 V— "i4>. Theorem 13. // b be any divisor of an ideal tit, and m = ttb, the number of integers of a complete residue system, mod ttt, which have with ttt the greatest common divisor b is <£(tt). Since by § I, ix, if the theorem be true for any particular residue system, mod m, it is true for all, we may take the system used in Th. 2. We have shown that the integers Bv lf $v 2 , •••,8vn[n], 4) where 8 is a number of b such that (8)/b is prime to n, and Vi,v 2 > •••>i'ni>] is a complete residue system, mod tt, comprise all and only those integers of a complete residue system, mod tn, which are divisible by b. Hence the integers of the complete residue system, mod m, which have with m the greatest common divisor b, are those of the system 4) in which the coefficient of b is prime to tt, and these are <£(tt) in number. Theorem 14. // b 1? b 2 , "- f h n be the different divisors of ttt, then I>(b,.) = «[m] Let b x , b 2 , •••,b„ be the different divisors of m, including ttt and (1). Then ttt = Xtl 1 '0 1 = tlt 2 b 2 = ••••= ttt n b«. Let Pi>P*> ••»/*»[«] 5) be a complete residue system, mod ttt, and separate these numbers CONGRUENCES WHOSE MODULI ARE IDEALS. 363 into as many classes as there are different divisors of m, putting into one class the <f>(m^) numbers that have with m the greatest common divisor b x (Th. 13), into another, the <f>(m 2 ) numbers that have with m the greatest common divisor b 2 , etc. It is evi- dent that each of the numbers 5) will.be in one and but one of these classes, and hence, since they are w[m] in number, *0»i) + *(**) H h*(n») =n[mj. But m lt m 2 , •••,m n are the different divisors of m, though in a different order from that of the b's. The theorem is therefore proved. The proof here given of this theorem is, it will be observed, dependent only upon Th. 13. The property of the 0-function thus shown completely defines the function and we shall be able to derive from it, as in R, the general expression for 0(m). From the general expression for 0(m) may then be obtained Th. II. We may also obtain Th. 14 from the general expression for 0(tn), as in R. These two proofs are left to the reader. Theorem 15. // m be any ideal other than (r), whose prime factors are p 15 p 2 , ••-,£»•, and b any divisor of m other than m, and if we separate all ideals of the form m no p being repeated, into two classes, I and II, putting in class I those such that m is divided by none or by the product of an even number of the p's, and in class II those such that m is divided by the product of an odd number of the p's, then exactly as many ideals of the one class are divisible by b as of the other. We see that the positive and negative terms of the developed product 1 in (-k)(-s)-(-k) coincide respectively with the ideals of classes I and II ; that is, 1 No meaning of addition or subtraction is to be abscribed to the + or — sign attached to these terms, it being simply observed that all the terms in the developed product are ideals, to some of which the sign + is attached and to others the sign — . 364 CONGRUENCES WHOSE MODULI ARE IDEALS. denoting by 2,m lt Stn 2 respectively the sums 1 of the ideals of these classes, we have m (-*)(-*)■••(-*)-*"■-*" • Let m = p 1 ei J) 2 e2 ---pr er . We shall prove the theorem first for the case in which e x =s*3 == • • • ss*e r =5 1 ; that is, m is not divisible by a higher power than the first of any prime ideal. Put Pip2-"Pr = a. We have K , -0( , --s)*--( , --s)-^- ,) *r ,) "-^ = 2^ — 2a 2 , where 5a lf 2a 2 have meanings corresponding to those of ^m^, 2m 2 . If now b be any divisor of a other than a, the number of a x terms which are divisible by h is exactly equal to the number of a 2 terms which are divisible by ft ; for, if we put a = &9i92 •*• 9«» where 9i, 9 2 > ,, *>9« are those prime factors of a which do not divide b, then the a/s and a 2 's, which are divisible by & are respectively the positive and negative terms of the developed product KSi— i)(?2— •'• (9.— I)? 6) Moreover, since &=H°> there is at least one prime ideal which divides a but not & ; that is, there is at least one g. Hence there are always exactly as many positive as negative terms in the developed product 6), and consequently as many a/s 1 This sum is to be understood in a purely formal sense as merely the aggregate of the ideals of the class connected by + sig ns > an d has, of course, no connection with the notation for the greatest common divisor given on p. 311. CONGRUENCES WHOSE MODULI ARE IDEALS. 365 as a 2 's divisible by B. The theorem is therefore proved when m is not divisible by a higher power than the first of any prime ideal. We proceed now to prove the theorem for the general case. Letting a, a lf a 2 retain the meaning assigned above, we have and it is evident that the ideals m lf m 2 coincide respectively with the products na t , tta 2 .. Let now b be any divisor of m other than m and let g be the greatest common divisor of the two ideals b = gb, and n = gc. We see that 6 is a divisor of a, for c is prime to b, and ca is divisible by b, since ca _ gca _ rta _ m ~F~~~g¥~ b~~ b"' " and m is divisible by b. From 7) it follows, since c is prime to b, that, if b = m, then c=(i) and b = a. Conversely, if b = a, and hence is divisible by all prime factors of m, then c, since it is a divisor of m but prime to b, must be ( 1 ) and hence b = tit. Excluding therefore the case b = m, so that we have always b=f=a, there are among the ideals a x exactly as many that are divisible by b as there are among the ideals a 2 . Since, moreover, the necessary and sufficient condition that an ideal nti = ncti = gca^ or ttt 2 = tta 2 = gca 2 , shall be divisible by b, = gb, is that a ± or a 2 shall be divisible by b, there are exactly as many of the ideals m 1 divisible by b as of the ideals m 2 . The theorem is therefore proved. This theorem and proof is interesting as illustrating once more how exactly everything concerning rational integers that involves no property other than that of divisibility, can be carried over to the general realm in terms of ideals. 366 CONGRUENCES WHOSE MODULI ARE IDEALS. As in the case of rational integers, the following theorem can be deduced from the one just proved. Theorem 16. a. If /(m) and F(m) be two functions of any ideal m that are connected by the relation S/(b)=F(m), 8) where b runs through all divisors of m, including m, then /(m)=2F(m 1 )— ^(m 2 ), 9) where m lf m 2 run through the values defined in the last theorem. b. If f(m) and F(m) be connected by the relation n/(b)=ILF(m), 10) then /(m) = TTZ7 ; \ . 11) To prove a) it is sufficient to observe that, if b x be any divisor of m other than in, it is a divisor of exactly as many of the m/s as of the m 2 's (Th. 15), and hence when in 9) we replace the F's by their values in terms of the /'s from 8), /(b^ will occur exactly as often with the plus sign as with the minus sign. Hence all terms in the second member of 9) will cancel with the excep- tion of /(m), which occurs but once. The proof of b) is similar and will be left to the reader. From Th. 16, a, we can easily obtain by the aid of Th. 14 the general expression for <£(m). From Th. 14 we have 2</,(b)=4m], where b runs through all divisors of m. Applying Th. 16, a, we have • /(m)=<£(m), F(m)=n[m], and hence <£(m) =2»[m 1 ] — 2«[m 2 ]. Since, moreover, 2l n 1 - 2l „ 2 = m( I -i-)( I -l)...(.-l) ( CONGRUENCES WHOSE MODULI ARE IDEALS. 367 and, if m m then [tnj = W>, • • • Pi «[m] we have 24mi ]- 24 m 2 ]-«[m]( I -^ ;j )(«-^ J )...( I -^ I ) and hence ^) = «[m]( I -^ ] )( I -^ J )...( I -^ ] ). Summing up what has been learned concerning the ^-function for ideals, we see that, exactly as in the case of the corresponding function in R, the function possesses the two properties : i. <f>(ah)=<f>(a)'4>(b) where a is prime to b. ii. 2<£(b) =w[m], where b runs through all divisors of m; and that either one of these properties completely defines the function, and from it may be deduced the general expression for <£(m) and the other properties, or we may as in R derive the general expres- sion for the function directly from its definition, and then from it get i and ii. The conception of $- functions of higher order and the theorems relating to them which hold for rational integers (Chap. Ill, § 6) can be at once extended to ideals. § 9. Residue Systems Formed by Multiplying the Numbers of a Given System by an Integer Prime to the Modulus. Theorem 17. // fi lf /x 2 , »*->/*»£«] be a complete residue system, mod m, and a any integer prime to trt, then a^a^, •••,a/>in[m] is also a complete residue system, mod m. The integers afi^a^, •••,«/*«[,„] are incongruent each to each, mod m, for from a.fii = a(jLj, mod m 368 CONGRUENCES WHOSE MODULI ARE IDEALS. it would follow that, since a is prime to m, fxi^=fij, mod m, which is contrary to the hypothesis that fi lt /x 2 , •••,/An[ m ] form a complete residue system, mod m. The integers a/^, a/x 2 , • • • , a/A n[ ] are, moreover, n[m] in number. They form, therefore, a com- plete residue system, mod m. Cor. // p 1} p 2 , •••,£$(„,) be a reduced residue system, mod m, and a be prime to m, then ap 1} ap 2 , •••,OLp <t>(m) is also a reduced residue system, mod m; for ap x ,ap 2 , •••,ap^ (m) are incongrueht each to each, mod m, prime to m, and <£(m) in number. Ex. Since 1, 2, 3, V^Ts. * + V^5> 2 + \/=~S, W^S, 1 + zV^, 2 + 2V — 5 constitute a complete residue system, mod (3), and V — 5 is prime to (3), V^S, 2V ==r 5, 3\/^5, — 5, — 5 + V^5, — 5 + tV^S* — 10, — 10 + V — 5, — 10 + 2V — 5 is also a complete residue system, mod (3). Likewise since 1, 2, V — 5, 2.\J — 5 is a reduced residue system, mod (3), V — 5, 2\/ — 5, — 5, — 10 is also a reduced residue system, mod (3). If p be any prime ideal and a an integer prime to p, it is evident from the above that there exists an integer a x such that 0^ = 1, mod p. We call a t the reciprocal of a, mod p. § 10. The Analogue for Ideals of Fermat's Theorem. The following theorem is for ideals the exact analogue of what Fermat's Theorem, as generalized by Euler, is for rational integers. The similarity in the proofs of the two theorems should be noticed. Theorem 18. If m be any ideal and a any integer prime to m, then a* (m) = 1, mod m. Let pitPts •••>?♦(*) be a reduced residue system, mod m. Then, since a Pl ,ap 2 , •••,ap <Km) 2) is also a reduced residue system, mod m, each number of 2) is CONGRUENCES WHOSE MODULI ARE IDEALS. 369 congruent, mod m, to some number of I ) ; that is, a Pi ■ Pji ] a Pi = Pj-2 Lmodm, 3) a P4>{m) — Pj^m) -> where P. , P; , ' ' ', Qi ± , are the numbers i), though perhaps in a different order. Multiplying the congruences 3) together, we have «* (w) -PiP* ' ' • P*(«) - P/, fe ' * * <W mod m ' from which, since Pl p 2 --p Mm) is prime to m, it follows that, a *(«) ssl> mo d m. Ex. Let m = (3 + V*^), and a = 3. We see that (3) is prime to (3 + V^5) and that 0(3 + V^ 1 -!) = 6; whence 3 6 =i, mod (3 + V^ r 5)> for 3 6 — 1, =728, is divisible by «[(3+ V — 5)L = H, and hence by Cor. 1. If p be a prime ideal, and a an integer not divisible by p, then a nm_1 = i, mod p. This is the exact analogue of Fermat's Theorem for rational integers Cor. 2. // p be any prime ideal, and a any integer, then a nM =a, mod p. § 11. Congruences of Condition. Just as in the rational realm we have so far considered con- gruences that may be compared to algebraic identities, the values of all the quantities involved being given and the congruences expressing simply the fact that the difference of the two num- bers is a number of the ideal that is the modulus, or, in other words, this difference considered as a principal ideal is divisible by the modulus. We shall now, as in the rational realm, consider congruences that hold only when special values are given to certain of the 24 370 CONGRUENCES WHOSE MODULI ARE IDEALS. quantities ; that is, the values of these " unknown " quantities are to be determined by the condition imposed by the congruence. To develop the theory of congruences of condition for ideal moduli it is necessary to introduce the conception of the con- gruence of two polynomials with respect to an ideal modulus; thus, If f(x lf x 2 , • -,x n ) be a polynomial in the n undetermined quan- tities x lf x 2 , "-,x n with coefficients which are integers of k(-\/m) and m be any ideal of &(\/m), we say that f(x x ,x 2 , ••■,x n ) is identically congruent to o with respect to the modulus m, if all its coefficients be divisible by m. 1 This relation is expressed symbolically by f(x x ,x 2 , >-,x n ) =o, mod m. Two polynomials, f(x t , x 2 , •••, x n ) and <j>(x lf x 2 , ..'., x n ), are said to be identically congruent to each other, mod m, if their difference be identically congruent to o, mod m, or, what is the same thing, if the coefficients of corresponding terms in the two polynomials be congruent, mod m ; that is, in symbols f(x 1 ,x 2 ,---,x n )=<l>(x 1 ,x 2 ,---,x n ), mod m, */ f(*v x 2, - - •? **) — 4> (*u *2> • • •» *"») ■■ °> mod nt. For example ; we have ( i + 3 V 3 ^)^ 2 + $*y + 7y 2 + 1 + 2 v^^ (8 + 3V =r 5)^ 2 +(2— V=5)*y + 2, mod (7, 3 +V=T). If f(x 1 ,x 2 ,---,x n )^<i>(x 1 ,x 2 ,-'-,x n ), mod m, i) and a 19 a 2i ••• f O« be any n integers of the realm, then evidently f(a 19 a 2 , •••,a n )=</>(a 1 ,a 2 , •••,a»), mod m. 2) If, however, 1) does not hold, then 2) does not hold in general 2 for every set of integers a t ,a 2 , ••• 9 a». 1 It will be understood throughout this discussion that the coefficients of a polynomial are integers of some certain quadratic realm and that the modulus is an ideal of this realm. 2 For an exception see § 13. CONGRUENCES WHOSE MODULI ARE IDEALS. 37 1 The demand that # 1$ x s , --,x n shall have such values and only- such that 2) will hold is expressed by writing f(x lf x 2f ■ -■,x n )==cf>(x 1 ,x 2 , ••-,*„), mod m. 3) Any set of integers satisfying 2) is called a solution of 3). The determination of all such sets, or the proof that none exists, is called solving the congruence 3). We call 3) a congruence of condition. If a lf a 2 , • • • , a n and p l ,p 2 ,-- m ,pn be two sets of n integers » each and a 2 ^p 2 a n = pn , mod m, 4) then by § 1, v, f (<*!,<*» -~, <*n) =f(P 1 ,P 2 ,~-,pn), mod m, and <j>(a 1 ,a 2 ,--,a n )=<l>(p 1 ,p 2 ,'-',Pn), mod m. Hence if a 1} a 2 , •••,#» be a solution of 3), p x ,p 2 , '-,Pn is also a solution. Two solutions so related are, however, looked upon as identical. In order that two solutions be different it is neces- sary and sufficient that the n relations 4) shall not hold simul- taneously. It is evident from the above that in order to solve any con- gruence, as 3), it is sufficient to substitute for the unknowns the (w[m] ) n sets of values obtained by putting for each unknown the w[m] numbers of a complete residue system, mod m, and observe which values of f(x x ,x 2 , •••,«#») so obtained are congruent to the corresponding values of <j>(x 1} x 2 , "-,x n ), mod m. There being only a finite number, (n[m]) n , of possible solu- tions, we can by this process always completely solve any given congruence. If the congruence have the form f(x lf x %9 • • •, x n ) == o, mod m, and a ls a 2 , •■•,a»bea solution, then f(x ± ,x 2 , --,x n ) is said to be zero, mod m, for these values of x 1} x 2 , '-',x n . 372 CONGRUENCES WHOSE MODULI ARE IDEALS. Ex. The solutions of the congruence (3 + V^^5)^ + ^ + 2 = o, mod (3, i + V — 5), are easily seen to be x = 1, y= — 1, and x m — 1, y = 1, mod (3, 1 + V — 5) • § 12. Equivalent Congruences. Two congruences, f ± (x lt x t9 -~,x n ) as f x (x % , #„•••,#»), mod m, i) $ 1 (x X9 Xt,-" f X n )ma<l» % (x 19 X 2 ,'",X n ) i mod m, 2) are said to be equivalent when every solution of the first is a solu- tion of the second and every solution of the second is a solution of the first. All that is said in Chap. Ill, § 10, regarding congruences in R applies equally to congruences with ideal moduli in any realm k( s/m). We have two transformations which lead to equivalent con- gruences; first, if 1 ) be the given congruence and F 1 (x 1 ,x 2 ,-'-,x n )^F 2 (x 1 ,x 2 ,--',x n ), mod m, 3) be any identical congruence, mod m, in x t ,x 2 , "-,x n , we can add 3) member by member to 1), obtaining AOi, *» ' ->*%) + Fi(*i, * 2 , ••*,*•)■■ /2O1, * 2 , • -;x n ) + ^2(^,^2, ••-,*„), mod m, a congruence equivalent to 1). By means of this transformation we can transpose any term with its sign changed, from one member of a congruence to the other and can thus reduce any congruence, as 1 ) , to an equivalent congruence of the form f{x u x v ..-,#») sso, mod m, whose second member is o. We shall hereafter assume the con- gruences with which we deal to have been reduced to this form. We may also by this transformation reduce the coefficients of f(x lf x 2f ---yXn) to their smallest possible absolute values, mod m, and thus lessen the labor of solving the congruences. In partic- CONGRUENCES WHOSE MODULI ARE IDEALS. 373 ular we can remove those terms whose coefficients are divisible by m. If m be such that a complete residue system, mod m, can be constructed entirely of rational integers, all coefficients of f(x Xi x 2 , • • •, x n ) can be replaced by rational integers. Using then this residue system for substitution the work becomes greatly simplified, especially when we remember that n[a] divisible by n[m] is a sufficient as well as necessary condition that a shall be divisible by m, if a be a rational integer. Ex. The congruence (4 + 3V = 5)^ 2 + (i— V ::: 5)^+(3 + 7V I=r 5)y 2 +i7 + 4V^5 = o, mod (7, 3 + \ /=r 5), 4) is equivalent to the congruence 2x 2 -f 4*y + 3/ + 5 = o, mod (7, 3 + >/—"$). This is equivalent to adding to 4) member by member the identical congruence (— 2 — 3\/ =r 5)* 2 + (3 + \/ z: S)*y — 7V^53> 2 — 12 — 4V— 5 = 0, mod (7, 3-f V^-S), —2 — 3V — S. 3 + V-—5i — 7V^ r 5, and —12 — 4V ^5 being all divisible by (7, 3 + V^ r 5)- A second transformation which leads to an equivalent con- gruence is the multiplication of both members of the congruence by any integer, a, prime to the modulus ; that is, the congruences f(x lt x 2i ••-,.*•„) =0, mod m, and af(x lt x 2 , • • • , x n ) = o, mod m, where a is prime to m, are equivalent. Conversely, we may divide all the coefficients of a congruence by any integer prime to the modulus, obtaining an equivalent congruence Ex. The congruences (3 + 3V^ r 5)* 2 + 9* — 6 — i5V^5=o, mod (3 + V" =r 5) > (i + V 3I 5)* 2 + 3* — 2 — sV^^o, mod (3 + V^S), are equivalent, since (3) is prime to (3 -f- V — 5). As a special case of the multiplication transformation, as we shall call the second of the above transformations, we have the 374 CONGRUENCES WHOSE MODULI ARE IDEALS. multiplication of the congruence by — I ; that is, the change of sign of each of its coefficients. § 13. Congruences in One Unknown with Ideal Moduli. The general congruence in one unknown has the form f(x) = a x n -f- a x x n ~ x -\ -f- 0*£E30, mod m, 1 ) where a ,a lf •••,a n are algebraic integers of any realm k, m an ideal of this realm, and n a positive rational integer. If p be an integer of k such that /(p) =0, mod m, p is called a root of 1 ) . The same analogies that existed in the rational realm in the case of congruences with one unknown when the modulus is a prime are easily seen to exist for prime ideal moduli, and their absence in the case of composite ideal moduli is equally marked. The reason is, of course, that just as in R the product of two integers is divisible by a prime number when and only when one of the integers is divisible by the prime, so the product of two integers, that is, two principal ideals, is divisible by a prime ideal when and only when one of the integers (that is, one of the prin- cipal ideals) is divisible by the prime ideal. Furthermore, we have the same difference in the case of congruences with prime ideal moduli between saying that all the coefficients are divisible by the modulus and that the congruence is satisfied by every value of the unknown; for example, as is easily seen from Fermat's Theorem as extended to ideals, the congruence #nm — ^ = 0, mod p, where p is a prime ideal, is satisfied by every integer of the realm, but its coefficients are not all divisible by p. Before taking up the general congruence in one unknown with ideal modulus, we shall consider that of the first degree. We give first two simple examples of congruences of higher degree. Ex. 1. Let (5 +\/=5)x*+ (1 +V-~5)* + 8 + 3V-5 = o,mod (3, 1 +V=5), 2) CONGRUENCES WHOSE MODULI ARE IDEALS. 375 be the given congruence. We observe first that I+V^SasO, mod (3, i + V^5)> 5 + V^5 = i, mod (3, i + V^5), 8 + 3V^~5 = 2, mod (3, i+V^5), and hence 2) reduces to x 2 + 2===o, mod (3, i + V^-5)- Substituting the numbers, 0, 1, 2, which constitute a complete residue system, mod (3, i + V — 5), we have 2 = 2hj=o, mod (3, I + V — 5). 1+2 = 3 = 0, mod (3, i + V - ^)- 4 + 2 = 6 = 0, mod (3, i + V^-5)- The congruence has therefore the two roots 1 and 2. Ex. 2. The congruence (5 — 6V^ r 5)* 2 + 7-r+i = o, mod (1 — V 37 ^. is equivalent to the congruence — jt-\-x+ 1 = 0, mod (1 — V"--5). 3) since 5 — 6\/^~5 = — I, mod (1 — V--5)i and 7ael, mod (1 — V^-S). Substituting the numbers 0, 1, 2, 3, 4, 5, of a complete residue system, mod (1 — V — 5)» m 3)» we see that the congruence has no roots. § 14. The General Congruence of First Degree with One Un- known. That there is always one and only one integer, £, of a complete residue system, mod m, that satisfies the congruence ax = /3, mod m, i) where a and (3 are integers, m any ideal and a prime to m, is evi- dent; for, if £ run through a complete residue system, mod m, then one and only one of the resulting products, a|i, is con- gruent to /?, mod m (Th. 17). Hence 1) has one and only one root, if. We proceed now with the discussion of the general con- gruence of the form 1), removing the restriction a prime to m. A necessary condition that the congruence shall have a solution is evidently, from (§ I, ix), that (3 shall be divisible by the 376 CONGRUENCES WHOSE MODULI ARE IDEALS. greatest common divisor, b, of a and m. We shall see that this condition is, as in the corresponding cases in R and k(i), also sufficient, and that, if it be satisfied, the congruence has exactly n[b] roots, incongruent, mod tn. To show this, let m = m 1 b, and take as a complete residue system, mod m, the w[m 1 ]n[b], = w[m], integers rr=i,2,...,n[b] where p is a number of m such that (p)/m is prime to b, and are complete residue systems with respect to the moduli b and m 1 respectively. We shall show that, if (/?) be divisible by the greatest common divisor of (a) and m, exactly w[b] of the numbers 2) satisfy 1). Let pfa-{- m De one °f tne integers 2). Since dp is divisible by m, we have by substitution in 1), as the necessary and sufficient condition that p8 h -f- /** shall satisfy 1 ) , ain = f$, mod m. But since (a)/b is prime to m, the numbers are all and only those numbers of a complete residue system, mod m, which are divisible by b (Th. 2). But ft is divisible by b. Hence there is one and only one of the integers 3) to which f$ is congruent, mod m. Let this integer be a^i. It is evident that of the integers 2) pK + /*i> p8 2 + /*i> •••,p8n[b] ~\- IH, satisfy the congruence 1), and are the only ones that do so. They are, moreover, »[b] in number. Hence the congruence 1) has exactly n[b] roots that are incongruent, mod m. CONGRUENCES WHOSE MODULI ARE IDEALS. 377 In particular, when b = ( I ) , that is, when a is prime to m, the congruence has, as we have already seen, one and only one root, all other integers satisfying it being congruent to this single one, mod m. In this case by means of Fermat's Theorem for ideals, we can find, as in the analogous case in the rational realm, a general expression for the root of the congruence a.v==j3, mod m, 4) where a is prime to m, and m is any ideal. Since a is prime to m, we have a* (m) ss 1, mod m, and hence or a/ta^" 0-1 = /?, mod m. Hence ^a* (m)_1 is the root of the congruence 4). The most obvious method of solving any given congruence, and one always applicable, is to substitute in turn the numbers of a complete residue system with respect to the modulus, thus deter- mining all the roots, if any exist, or proving the non-existence of a root. This is usually the easiest method when the norm of the modulus, m, is small, and especially when the numbers 1, •••,»[m] — 1 constitute a complete residue system, mod m. This method has already been used in § 13, Exs. 1 and 2. We shall further illustrate it and also the method depending on Fermat's Theorem on the congruence S^^i + V 13 ^ mod (7, 3 + V T=r 5), 5) The numbers o, 1, 2, 3, 4, 5, 6 constitute a complete residue system, mod (7> 3 + V — 5), substituting them in turn, we have 6) 7) 8) mod (7, 3 + V — 5), 9) 10) 11) 12) 0— (i + V — 5)= — i — V — 5#eO 5— (i + \/^~5) = 4 — V^Ssso 10— (1 + V^5) = 9 — V^H^o 15— (1 + V—^) = 14 — V^^ 20— (1 + V :i3 5) = 19 — V--5 45 25— (1 + y/^-5) =24 — V^he^o 30— (1 + v^s) =29 — V^Se^o 378 CONGRUENCES WHOSE MODULI ARE IDEALS. all of which results, except 7) and 12), follow at once from the fact that n[— 1 — V^5L =6, n[g — V^l], =86, «[i4 — V^L =201, wfig — V — 5], =366, and «[20, — V — 5], =846, are none of them divisible by n(7, 3 + V — 5), = 7, and hence none of the numbers, — 1 — V^5> 9 — V—l. 14— V--& 19 — V^S and 29 — \/~5 can be divisible by (7, 3 + V — 5)- To obtain 7), we observe that ^[4 — V — 5L =21, is divisible by »(7i 3 + V — 5). and therefore 4 — V — 5 may be divisible by (7, 3 + V — 5). This is seen to be the case since 7— (3 + V^S) = 4 — V^. Hence 1 is a root of 5). To obtain 11), we proceed exactly as with 7) and find that the condition w[24 — V — 5] divisible by n(7, 3 + V — 5)> which is necessary in order that 24 — V — 5 sna H De divisible by (7, 3 + V — 5), is satisfied, but that the equation 7x + (3 + V^-S)? = 24 — V^5 gives as values for x and y 27 *=7* /=— «• These not being both integral, 24 — V — 5 is not divisible by (7, 3+V — 5)- This last result could have been obtained also by showing that (7, 3 + V~=~5, 2 4 -V^5) = 0). This method is, in general, if a be any integer and b, = (&, /3,), any ideal, to show that a is not divisible by b, it is sufficient to show that the ideal (j8 lf )8 2 , a) contains a rational integer smaller than any in b. If we had noticed originally that, since 5= — (V — 5) 2 > and (7, 3+V — 5) is prime to V — 5, the congruence has one and only one root, the work, after finding that 1 was a root, would have been unnecessary. It was given in full to illustrate this most primitive but fundamental method of solution, which is entirely independent of the above discussion. We shall illustrate now upon the same congruence the method de- pendent upon Fermat's Theorem. Since 5 is prime to (7, 3 + V— ~S), and 0(7, 3 + V^ 17 ^) =6, we see that (1 + V — 5)5 8 is the root of 5). To show that (i + V" zr 5)5 5 =i, mod (7, 3 + V zr 5), we observe that i + V^ees — 2, mod (7, 3 + V^ : 5), and 5 = — 2, mod (7, 3 + V^5), and hence (i + V Tr 5)5 C E=(-2)(-2) 5 ==6 4 =i, mod (7, 3 + V^5). The solution of a congruence of the form 1) where a is not prime to m is perhaps most conveniently accomplished by means of the method sug- gested by the general discussion of this case. We shall illustrate this by two examples. CONGRUENCES WHOSE MODULI ARE IDEALS. 379 Ex. 2. 2x = ~, mod (i + V^^)- The greatest common divisor of (2) and (i + V — 5) is seen to be (2, 1 + V — 5), that does not divide (7). Hence the congruence has no root. Ex. 3. 2.r==i — V — 5, mod (i + V — 5). 13) Since (1 — V — 5) is divisible by (2, 1 + V — 5), the greatest common divisor of (2). and (1 +V — 5), the congruence has 11(2, 1 +V — 5), =2, roots. We have (3.1 + V-5). (2, i + V-5) Taking as a complete residue system, mod (3, i + V — 5), the num- bers o, 1, 2, and substituting these numbers in 13), we have o— (1 — V=S)=— 1+ V^5 + o " 2 — (1 — V~5> = I + V=S»o 4— (1 — <=$) = 3 + >T=5 + o We have therefore, in the notation of the general discussion, W= 1, mod (1+ V — 5). p., mod (1 + V — 5). Since (3) (3, I4-V-5 . =(3, is prime to (1 + V — 5), we may take p = 3, and since o, 1 constitutes a complete residue system, mod (2, i + V — 5), we have as the two roots of 13) 3.0+1 = 1, and 3-1 + 1=4. The reader may verify these results, as found in examples 2 and 3, by direct substitution of the numbers of a complete residue system, mod (i + V^5). These two congruences (Exs. 2 and 3) will serve as instructive ex- amples of the dependence of the entire theory of algebraic numbers upon the unique factorization theorem, and the necessity for the introduction of the ideal. In Ex. 2, 2 and 1 + V — 5> considered merely as integers of k(y/ — 5), are prime to each other, and, were it not for the failure of the unique factorization theorem in k(yj — 5), we should expect the congruence therefore to have a single root in accordance with the results obtained in R and k(i). Substituting the numbers of a complete residue system, mod i + V — 5, we find that it has no root. Likewise in Ex. 3, considering the numbers involved merely as integers of £(V — 5)> we should expect the congruence to have a single root. 380 CONGRUENCES WHOSE MODULI ARE IDEALS. Substituting the numbers of a complete residue system, mod 1 + V — 5» we find that it has two roots. The reason for these discrepancies is made plain when we resolve 2, 7, 1 — V — S» J + V — 5, into their prime ideal factors. § 15. Divisibility of one Polynomial by another with re- spect to a Prime Ideal Modulus. Common Divisors. Common Multiples. If p be any prime ideal of a realm k, we have the following definition : A polynomial, f(x), is said to be divisible with respect to the modulus p by a polynomial <f>(x), when there exists a polynomial Q(x) such that fi*\mQ(j*)+{x) l mod p. We say that <f>(x) and Q(x) are divisors or factors, mod p, of f(x), and that f(x) is a multiple, mod p, of <£(•*") an d Q(x). The sum of the degrees of the factors of f(x) is evidently equal to the degree of f(x). The coefficients of f(x), <K#) and Q{x) are understood to be in- tegers of k. Ex. It is easily seen that (4 + 3V"= r 5)^-^ + ^ + V=l^+(i + V"=l)^ + 2 = (V-5^ 2 +( I +V :==r 5)^+2)((3+2V =r 5)^+i), mod (7, S+V^JO'* Hence V= r 5^ 2 +(i + V :=:r 5)^ + 2 and (3 + 2 V'=5)* a + 1 are divisors, mod (7, 3 + -y/ — 5), of We have the same consequences of this definition and the same definitions of common divisor and common multiple for prime ideal moduli as for rational prime numbers (Chap. Ill, § 15). § 16. Unit and Associated Polynomials with respect to a Prime Ideal Modulus. Primary Polynomials. We see as in the rational realm that the integers of the realm, not divisible by p, divide every polynomial with respect to the CONGRUENCES WHOSE MODULI ARE IDEALS. 38 I modulus p, since they divide i, mod p, and that these are the only polynomials having this property. We call therefore the integers of k, which are not divisible by p, the unit polynomials, mod p, or briefly the units, mod p. Since two polynomials that are congruent, mod p, are consid- ered as identical, we can take as the units, mod p, the integers of any reduced residue system, mod p. Two polynomials which differ only by a unit factor, mod p, are called associated polynomials and are looked upon as identical in all questions of divisibility, mod p. Two polynomials that are associated with a third polynomial, mod p, are associated with each other, mod p. Two polynomials that are associated, mod p, are evidently of the same degree and each is a divisor, mod p, of the other. Conversely, if two polynomials be each divisible, mod p, by the other, they are associated, mod p. Two polynomials that have no common factor, mod p, other than the units, are said to be prime to each other, mod p. Any polynomial, f{x), has n(p) — I associates, mod p. Of these, one and only one has the coefficient of its highest degree i. This one is called the primary associate, mod p of f(x). For example, the six polynomials x* _|_ 2.v — 3, 2x 3 -\- 4X — 6, 3,r 3 -\- 6x — 2, 4. r 3_L. x—s, 5* 3 + 3-r— 1, 6;r 3 + 5* — 4, are associated, mod 7, and x 3 -\-2x — 3 is the primary one. § 17. Prime Polynomials with respect to a Prime Ideal Modulus. Determination of the Prime Polynomials, mod p, of any Given Degree. A polynomial that is not a unit, mod p, and that has no divisors, mod p, other than its associates and the units, is called a prime polynomial, mod p. If it has divisors, mod p, other than these it is said to be composite, mod p. We can determine the primary prime polynomials, mod p, of any given degree, n, by the process employed in the same case in 382 CONGRUENCES WHOSE MODULI ARE IDEALS. the rational realm; that is, write down all primary polynomials, mod p, of degree n; then, having determined by multiplying together the primary polynomials, mod p, of degree less than n, all composite primary polynomials, mod p, of degree n, we strike them from the list of all primary polynomials, mod p, of degree n. Those left are evidently the primary polynomials, mod p, of degree n. § 18. Division of one Polynomial by another with respect to a Prime Ideal Modulus. Theorem 19. // f{x) be any polynomial and <f>(x) be any polynomial not identically congruent to 0, mod p, there exists a polynomial Q(x), such that the polynomial f(x) — Q(x)<f>(x)==R(x), mod p, is of lower degree than <j>(x). The operation of determining the polynomials Q{x) and R(x) is called dividing f(x) by 4>(x), mod p. We call Q(x) the quo- tient and R(x) the remainder. The proof of this theorem is pre- cisely the same as that for the corresponding one in the rational realm. The conception of the congruence of two polynomials with respect to a double modulus is the same for a prime ideal as for a rational prime number. § 19. Unique Factorization Theorem for Polynomials with respect to a Prime Ideal Modulus. We shall now show that, just as a polynomial whose coefficients are rational integers can be resolved in one and but one way into prime factors with respect to a rational prime modulus, so a polynomial, whose coefficients are integers of any given quadratic 1 realm, can be resolved in one and but one way into prime factors with respect to a prime ideal modulus. The proof will be seen to be identical with that employed for rational numbers. We begin by stating the following theorem, whose truth is evident. 1 This holds for realms of any degree. CONGRUENCES WHOSE MODULI ARE IDEALS. 383 Theorem 20. // f(x) ==Q(x)<f>(x) + R(x), mod p, every polynomial that divides, mod p, both f(x) and <f>(x) divides both tf>(x) and R(x) and vice versa; that is, the common divisors, mod p, of f(x) and <f>(x) are identical with the common divisors, mod p, of 4>{x) and R(x). Theorem 21. // f 1 (x), f 2 (x) be any two polynomials and p a prime ideal, there exists a common divisor D(x), mod p, of fi( x )y fz( x )> such th Qt D( x ) W divisible, mod p, by every com- mon divisor, mod p, of f x (x), f 2 (x), and there exist two poly- nomials, <\>x{x), <£ 2 (- r )> such that AWfcW +/iW*W"^W« mod P- We may evidently assume f 2 (x) of degree not higher than f x (x). Dividing f x (x) by f 2 (x), mod p, we can find two poly- nomials, Q 1 (x), f s (x), such that /,(*) BE &(*)/,(*) + /,(*), mod p, f 3 (x) being of lower degree than / 2 (.r). Dividing f 2 (x) by / 3 (.r), mod p, we have /.(*)■&(*)/,(*)+/,(*), mod p, where f t (x) is of lower degree than / 3 (.r), and similarly /,(f) = 0.(*)/«Or)+/.(f) - /„. 2 (.r) = Q B . 2 ('.r)/„_ 1 (.v)+/„(^) -. mod p, /*.,<*) «a„(*)/.(*) a chain of identical congruences in which we must after a finite number of steps reach one in which the remainder, f n+1 (x), is o, mod p, since the degrees of that remainder continually decrease. By Th. 20 the common divisors, mod p, of f n (x) and fn-^x) are identical with those of /„_ 1 (^r) and f n - 2 (x), those of /n_i(-^), fn- 2 (x) with those of f n . 2 (x), f n - 3 (x), and finally those of f 3 (x), f 2 (x) with those of f 2 (x), f x {x). But f n (x) is a common divisor, mod p, of f n (x) and fn-i(x) and is evidently divisible by every common divisor of f n (x) 384 CONGRUENCES WHOSE MODULI ARE IDEALS. and fn-i(x)- Hence f n (x) is the desired common divisor, D(x), mod p, of f ± (x) and f 2 (x). If now we substitute the value of f 3 (x) in terms of f t (x), f 2 (x), obtained from the first of these congruences, in the second and the values of f 3 (x) and f 4 (x) in terms of f t (x), f 2 {x) in the third and continue this process until the congruence fn- 2 {x)=Q n _ 2 {x)f n _ 1 {x) +/»(.*•), mod p, is reached, we shall obtain a congruence, fi(*)+i{*)+M*)+Mi*)mD{s), mod p, where faix), <f> 2 (x) are polynomials. Cor. If f 1 (x), f 2 (x) be two polynomials prime to each other, mod p, there exist two polynomials, <j> x {x), <)> 2 (x), such that /iO)4>i<» +/iWfcW.*!i mod p. In this case D (x) is an integer, a, not divisible by p, and we have /iO)$iO) +f$x)* M (x)"mCh mod p, whence, multiplying by the reciprocal of a, mod p, we obtain f x {x)4> 1 {x)+f 2 {x)4> 2 {x) = i, mod p. Theorem 22. // the product of tzvo polynomials, f x (x), f 2 (x) , be divisible, mod p, by a prime polynomial, P(x), at least one of the polynomials f t (x), f 2 {x) is divisible, mod p, by P(x). Let f 1 (x)f 2 (x)^Q(x)P(x),modp, 1) where Q(x) is a polynomial, and assume f x (x) not divisible, mod p,byP(x). Then f x (x) and P(x) are prime, mod p, to each other and by Th. 21, Cor. there exist two polynomials, ^(.r), <f> 2 (x), such that /1OH1O) +P(s)+t{x)mi $ mod p. 2) Multiplying 2) by f 2 {x) and making use of 1), we have P(*j[Q(*)+i(*) +&(*)+*(«)]■/,(*)) mod fc where Q(x)<}> 1 (x) -\-f 2 (^)<f> 2 (x) is a polynomial. Hence f 2 (x) is divisible, mod p, by P(^). CONGRUENCES WHOSE MODULI ARE IDEALS. 385 Cor. 1. // the product of any number of polynomials be divis- ible, mod p, by a prime polynomial, P{x), at least one of the polynomials is divisible, mod p, by P(x). Cor. 2. // neither of two polynomials be divisible, mod p, by a prime polynomial, P(x), their product is not divisible, mod p, by P(x). Theorem 23. A polynomial, f(x), can be resolved, mod p, in one and but one way into a product of prime polynomials, mod p. The proof of this theorem is identical with the corresponding one in the rational realm. We can now evidently write any polynomial, f(x), in the form fWaaCAW)^?,^))*- (Pn(*)) en , mod p, where P x (x), P 2 (x), •••, P n (x) are the unassociated prime fac- tors, mod p, of f(x). If we take P 1 (x), P 2 (x), •••, P n (x) primary, the resolution is* absolutely unique. The representations of the greatest common divisor and least common multiple,* mod p, of two polynomials are identical with those in the rational realm. The resolution of any polynomial into its prime factors, mod 1p, may be effected by the method employed in the case of rational numbers. § 20. The General Congruence of the nth Degree in One Un- known and with Prime Ideal Modulus. Theorem 24. If p be a root of the congruence f(x)=a x n + a x x n - 1 -\ f-a„ = o, mod p, 1) f(x) is divisible, mod p, by x — p, and conversely, if f{x) be divisible, mod p, by x — p, p is a root of 1). Dividing, mod p, f(x) by x — p, we have f(x) = (x — P )<p(x)+R(p), mod p, whence, since p is a root of 1), f(x) = (x — p)<p(x), mod p; 25 386 CONGRUENCES WHOSE MODULI ARE IDEALS. that is, f(x) is divisible, mod p, by x — p. The converse is evident. If f(x) be prime, mod p f the congruence i) evidently has no roots. The converse is, however, not true; that is, f(x) may be composite, mod p, but i) have no roots, for the prime factors of f(x), mod p, may all be of higher degree than the first. This theorem gives us another method for determining the factors, mod p, of the first degree of any polynomial in x. Some of these factors may be alike and we are led therefore to say that p is a multiple root of order e of i), if f(x) be divisible, mod p, by (x — p) e but not by (x — p) e+1 . If, therefore, p 1 ,p 2 ,'",p m be the incongruent roots of i) of orders e lt e 2 , . . • , e m respectively, we have f(x)m(x— pt)H*— p 2 )*--- (*—f>m) em fi(*), mod p, where f t (x) is a polynomial having no linear factors, mod p, and whose degree ^ is such that where n is the degree of f(x). Counting a multiple root of order e as e roots, we see that i) has exactly as many roots as f(x) has linear factors, mod p, and have the following important theorem : Theorem 25. The number of roots of the congruence f(x)=a x n -\-a 1 x n - 1 -{ \-a n = o, mod p, where p is a prime ideal, is not greater than its degree. Cor. 1. // the number of incongruent roots of a congruence with prime ideal modulus be greater than its degree, the con- gruence is an identical one. Cor. 2. // the congruence f(x) =0, mod p, 2) have exactly as many roots as its degree, and <f>(x) be a divisor, mod p, of f(x), then the congruence <f>(x) =0, mod p, CONGRUENCES WHOSE MODULI ARE IDEALS. 387 has exactly as many roots as its degree; for /(*)M+(*)0(*)> m od p, where Q(x) is a polynomial in x, and every root of the con- gruence 2) is a root of either the congruence <f>(x)=o, mod p, 3) or of the congruence Q(x) =0, mod p. 4) Moreover, the sum of the degrees of 3) and 4) is equal to the degree of 2). If, therefore, <f>(x) had fewer roots than its degree, then Q(x) must have more roots than its degree, which is impossible. Hence the corollary. § 21. The Congruence x* (m) — 1 = 0, mod m. Although in the case of congruences of degree higher than the first the theorem just given tells all that we can in general say regarding the number of the roots, still there is, as in the rational realm, one important case in which the number of roots is always exactly equal to the degree of the congruence. Theorem 26. The congruence 4r* (a) ss 1, mod m, 1) has exactly </>(m) roots. The 4>(m) integers of a reduced residue system, mod m, evi- dently satisfy 1). Moreover, since by §1, ix two integers con- gruent, mod m, have with m the same greatest common divisor and the greatest common divisor of (1) and m is (1), every root of 1) must have with m the greatest common divisor (1) ; that is, be prime to m. Hence the number of roots of 1 ) is exactly equal to <£(m), its degree. Ex. 1. The congruence jf^+i^BM, mod(i+ 1 /=5), or * 2 == 1, mod (1 -f V -~5)> has two roots, 1 and 5, Likewise the congruence **CW"=*>- 1, mod (7, 3 + /=5), 388 CONGRUENCES WHOSE MODULI ARE IDEALS. or * 6 =ee i, mod (/, 3 + \f = -$), has six roots, i, 2, 3, 4, 5, 6. Ex. 2. Consider the congruence ^ (2l/ ^5, _5 + y— 5) ^ f> mod (2^=5, _ 5 + ^5), 2) Since (2V"^5, - 5 + V-^S) = ( V^l) (2, 1 + V^), we have 0(2V^-5> — S + V^) =0(V^5)^(2, 1 + V 3 ^) =4-1 = 4. Substituting therefore in the congruence **aasl, mod UV— ~S, — S + V"^). the numbers o, 1, 2, 3, 4, 5, 6, 7, 8, 9, which form a complete residue system, mod (2V — 5, — 5 + V — 5)* we see that the numbers 1, 3, 7, 9, which form a reduced residue system, mod (2\/ — 5, — 5 + V — 5), are the only ones which satisfy the congruence. Cor. If d be a positive divisor of $(p), the congruence x a — j == o, mod p, where p is a prime ideal, has exactly d roots. This follows at once from Th. 25, Cor. 2, since x d — 1 is a divisor, mod p, of x* M — 1. The congruence x nV ^ — ,r = o, mod p, having the n [p] roots pu P2> * ' •> p n ipi equal in number to its degree, we have the identical congruence x n W — x==(x — Pl )(x — P2 ) ... (x — PnM ), mod p. For example x 7 — x = x(x — i)(x — 2)0 — 3)0 — 4)0— 5)0" — 6), mod (7, 3+ V^S)- § 22. The Analogue for Ideals of Wilson's Theorem. The result just obtained gives us a proof of the following theorem : Theorem 27. If p be a prime ideal and p lf p 2 , •••, p^^) a reduced residue system, mod p, then P1P2 •••?,*,(»+ 1=0, mod p. Since the congruence X *M — 1=0, mod p, CONGRUENCES WHOSE MODULI ARE IDEALS. 389 has exactly <f>(p) roots, Pl ,p 2 , • • - t p^ Mt we have by § 21 x .*(») i==(x — Pl )(x — p 2 ) •■• (* — p* ( »), mod p. Putting jt = o, we have — I3s(- pi)(— p,) ••• (— p«»), mod p, whence, since <£(p) is even, except when n[p] = 2, P1P2 ~'P4M+ l —°> mod p, which evidently holds also when n[p] = 2. Ex. Let p = (7, 3 + V — 5) ; then 1, 2, 3, 4, 5, 6 is a reduced residue system, mod (7, 3 -f- V — S)j an d we have 1 -2 • 3 .4. 5 -6+1 = 721=0, mod (7, 3 + V— !)• § 23. Common Roots of Two Congruences. The common roots of two congruences f 1 (x)==o, mod p, and / 2 (.r)=o, mod p, are evidently the roots of the congruence <f>(x) sago, mod p, where <f>(x) is the greatest common divisor, mod p, of f t (x) and f 2 (x). Since the congruence 2*1*1 — .r==o, mod p, has for its roots the numbers of a complete residue system, mod p, the incongruent roots of any congruence f(x) =0, mod p, will be the roots of the congruence <p(x) =0, mod p, where <f>(x) is the greatest common divisor, mod p, of x nlx ^ — x and f(x). This gives us another method of determining all the incon- gruent roots of any given congruence with prime modulus. 390 CONGRUENCES WHOSE MODULI ARE IDEALS. § 24. Determination of the Multiple Roots of a Congruence with Prime Ideal Modulus. The multiple roots of the congruence /0)=o, mod p, 1) may be determined just as in the case of rational integers. Let P(x) be a prime polynomial, mod p, and let f(x) be divisible, mod p, by [P(x)] e but not by [P(x)] e+1 ; then f(x) = [P(x)Y<f>(x), mod p, or, what is the same thing, where F(x) and <f>(x) are polynomials in x, with coefficients which are integers of the realm k, to which p and the coefficients of f{x) belong, and F(x) is identically o, mod p. Differentiating 2), we have f(*)«=[P(*)J**[«P(£)#(jr) +P(,*)*'(*)] +F'(x), where P'(x), <f>'(x) and F'(x) are polynomials in x with coeffi- cients which are integers of k, and F'(x) is identically o, mod p, for all coefficients of F(x) being divisible by p, all coefficients of i 7 '^-) are divisible by p. Hence f(x)mlP{xy]- 1 i> x (x),taodp, where $ x (x) is a polynomial in x, with coefficients which are integers of k, and is, moreover, not divisible, mod p, by P(x), for *,(*) = «/»(*)+(*) +P(x)*'( J r), where P'(x) is of lower degree than P(x) and <#>(^) is prime, mod p, to P(^")- Therefore f(x) is divisible, mod p, by the prime factor P(x) exactly once less often than f(x). In particular, if f(x) be divisible, mod p, by (x — p) e but not by (x — p) e+1 , then f(x) is divisible, mod p, by (x — p) e_1 but not by (x — p) e . Hence the theorem : CONGRUENCES WHOSE MODULI ARE IDEALS. 39 1 Theorem 28. If the congruence f(x) =0, mod p, have a multiple root, p, of order e, the congruence f(x) =0, mod p, has the multiple root p of order e — 1. If the greatest common divisor, mod p, of f(x) and f(x) be 4>(x), then the roots of the congruence <£0)e=o, mod p, 3) if it have any, will be the multiple roots of 1) and each root of 3) will occur once oftener as a root of 1) than as a root of 3). It may happen, of course, that f(x) and f{x) have a common divisor, <f>(x), mod p, and yet 1) has no multiple roots. In this case the repeated prime factors, mod p, of f(x) are of degree higher than the first, and <j>(x), therefore, contains no factor of the first degree, mod p. § 25. Solution of Congruences in One Unknown and with Composite Modulus. The solution of a congruence of the form f(x) = a x n + a x x n ~ x -| +a„ = o, mod m, 1 ) where m = m 1 m 2 • • • m t , m 19 m 2 , ••■•, mt being ideals prime each to each, can be reduced to the solution of the series of t congruences f(x) =0, mod m lt ' f(x) ^o, mod m 2 , f(x) ^o, mod m 2) Every root of I ) is evidently a root of each of the congruences 2), and conversely any integer, p, of the realm which is simul- taneously a root of each of the congruences 2) is a root of 1), for if the integer f(p) be divisible by each of the ideals m lf m 2 , •••, mt, which are prime each to each, it is divisible by their product. 392 CONGRUENCES WHOSE MODULI ARE IDEALS. If therefore a lf a 2 ,---,a t be roots of the congruences 2) and p be chosen so that p = a 1} mod Bin ) P = a 2 , mod m 2 , I 3) pzz=a t , mod mt, then p is a root of 1). Since m 1} m 2 , ■■■,mt are prime each to each, it is by Th. 10 always possible to find p so as to satisfy the conditions 3). Let p 1 ,p2>'">Pt be auxiliary integers selected as in Th. 10; then p = aj$i + a 2 (3 2 H h atfitt mod m, 4) is a root of 1), and, if the congruences 2) have respectively ^2> "->h incongruent roots, then by Th. 10 1) has lj 2 ••• l t in- congruent roots, which are obtained by putting for a lt a 2 , •••,&* in 4) respectively the l lt l 2 , ••-,/* roots of the congruences 2). In particular, if any one of the congruences 2) have no root, then 1) has no root. We may now suppose m = p&pf* • • • pr er , where the }>'s are different prime ideals, and show, as in the corresponding case in R (p. 96), that the solution of the congruence f(x) ^o, mod p e , can be made to depend upon that of f(x) ^o, mod p^ 1 , and hence eventually upon that of /(jr)^o, mod p, the same method being applicable with slight modifi- cations. § 26. Residues of Powers for Ideal Moduli. // a be prime to the ideal m, and fi^a*, mod m, where t is a positive rational integer, /? is said to be a power residue of a with respect to the modulus m. For example, since — 2V— 5=(i+V— 5) 3 > mod (7, 3-f-V— 5), we say that — 2\/ — 5 is a power residue of 1 -|~V — 5> mod (7> 3 +V — 5)- Two power residues of a which are congruent, mod m, to each other and hence to the same power of a, are looked upon as the same. CONGRUENCES WHOSE MODULI ARE IDEALS. 393 A system of integers such that every power residue, mod m, of a is congruent, mod m, to one and only one integer of the system is called a complete system of power residues of a, mod m. These integers may evidently be selected from among the integers of any reduced residue system, mod m. The following table gives the power residues of all numbers of a reduced residue system, mod (7, 3+V — 5), the system taken being 1,2,3,4,5,6. m = (7, 3+ V a a 1 a 2 a 3 a* a 5 a 6 1 I I I I I I I 2 4 I 2 4 I I 3 2 6 4 5 I I 4 2 1 4 2 I I 5 4 6 2 3 I I 6 1 6 1 6 1 We ask now what is the smallest value, t a , of t, greater than o, for which a*= 1, mod m. That such a value of t always exists and is equal to or less than <£(m) is evident from Th. 10 by which we have, since a is prime to m, a* (m) s= 1, mod m. Giving to t a the above meaning, we say that the integer a apper- tains to the exponent t a with respect to the modulus m. We see, by consulting the above table, that 3 and 5 appertain to the exponent 6; that is, <£(m), mod (7, 3 +V — 5), that 2 and 4 appertain to the exponent 3, mod (7, 3+V — 5)» and that 6 appertains to the exponent 2, mod (7, 3+V — 5). It is evident that, if a = /?, mod m, then a and (3 appertain to the same exponent, mod m. Hence to find the exponents to which all integers appertain, mod m, it is only necessary to ex- amine the numbers of a reduced residue system, mod m. 394 CONGRUENCES WHOSE MODULI ARE IDEALS. Theorem 29. If the integer a appertain to the exponent 4, mod m, then the t a powers of a, it, a, a 8 ,...., a*.- 1 , 1) are incongruent each to each, mod m. Let a s , a 8+r be any two of the numbers 1). If a s+r = a s , mod m, 2) then, since a is prime to m, a r =i, mod m. 3) But r is less than t a and 3) is therefore impossible, since a appertains to t a . Hence 2) is impossible. Theorem 30. // a appertain to the exponent t a , mod m, any two powers of a with positive exponents are congruent or incon- gruent, mod m, according as their exponents are congruent or incongruent, mod t a . Let a* 1 , a 82 be any two powers of a, s lf s 2 being positive rational integers, and let *i = qJa + r i» s * = q*t a + r *> where q lf q 2 are positive rational integers and o^r 1 <t a , o^r 2 <t a , r x ^r 2 . 4) If otfi*«-H-i = a^- +r2 , mod m, 5) then a ri = a r2 , mod m, 6) and hence, since a is prime to m, a ri ~ r2 = 1, mod m. But from 4) we have o^r x — r 2 < t ai whence, since a apper- tains to t a , mod m, r 1 = r 2 . 7) Therefore s ± = s 2 , mod /<., 8) is a necessary condition that we shall have a si = a * 2j mo d m. 9) Moreover, from 8) follow in turn 7), 6) and 5). Hence 8) is also a sufficient condition for the existence of 9). We have therefore CONGRUENCES WHOSE MODULI ARE IDEALS. 395 a E=a' a+1 = a 2 ' a+1 = - ,ta-l -mod m; that is, the same law of periodicity holds for power residues with respect to ideal moduli as in the case of rational integers. This can be verified by an examination of the table (p. 393), where we see, for example, that 2 appertains to the exponent 3, mod (7, 3+V^lO, and that 2° mm 2 3 sg 2 6 mm mod (7, 3 +V :=: 5), 2 ==2* 2- = 2 l 2' = 2 8 = and = 3 = 6=--. 1=4 = 7=... 2^5 = 8=. --J mod 3. Theorem 31. The exponent, t a , to which an integer, a, apper- tains with respect to the modulus m, is always a divisor of <£(m). Since a+^ssissa , mod m, we have by Th. 30 <£(nt) =0, mod t a . Theorem 32. If two integers, a lt a 2 , appertain, mod tn, to two exponents, t lt t 2 , which are prime to each other, then their product, a^a 2 , appertains, mod m, to the exponent, t x t 2 . Let a x a 2 appertain to the exponent t, then (a 1 o 2 ) # ssi, mod m. 10) Raising both members of 10) to the ^th power, we have a^hta^* mm i, mod m. But a^^mmi, mod m, and hence a^^i, mod tn. Therefore, since a 2 appertains to the exponent t 2 , mod m, t x t must be a multiple of t 2 , whence, since t lf t 2 are prime to each other, it follows that t is a multiple of t 2 . 396 CONGRUENCES WHOSE MODULI ARE IDEALS. In like manner we can show that Ms a multiple of t v Therefore, t being a multiple of both t 1 and t 2 , is a multiple of their product, t x t 2 . Hence the smallest possible value of t for which 1 ) holds is f x # 8 . Therefore, a x a 2 appertains to the exponent ^ 2 , mod m. Ex. We see from the table (p. 393) that 2 and 6 appertain, mod (7, 3 + V — 5)> respectively to the exponents 3 and 2, and that their product, 12, ^5, mod (7, 3 + V — 5), appertains to the exponent 6, mod (7, Limiting ourselves now to the case in which the modulus is a prime ideal p, we ask whether there are integers appertaining to every positive divisor of <j>(p), an d, if so, how many? An examination of the table will show us how matters stand when p = (7, 3 +V =r 5)- We have <f>(y, 3 -f-V — 5) =6, and the positive divisors of 6 are 1, 2, 3 and 6. To 1 appertains the single integer I. To 2 appertains the single integer 6. To 3 appertain two integers, 2 and 4. To 6 appertain two integers, 3 and 5. Theorem 33. To every positive divisor, t, of <f>(p) there appertain <f>(t) integers with respect to the modulus p. Assume that to every positive divisor, t, of <f>(p) there apper- tains at least one integer, a. We shall show that, if this assump- tion be true, there appertain to t <f>(t) integers; that is, to every positive divisor, t, of <f>(p) there appertains either ^(t) 1 integers or no integer. Let \J/(t) denote the number of integers appertaining to t. Each of the integers a°=i,a,a 2 , •••,a* _1 , 11) is a root of the congruence £*■■ 1, mod p; 12) for, if a r be any one of these integers, then (ar) t =(a t ) r Bsi i mod p, 1 We consider t simply as a rational integer, and <t>{t) is to be understood in this sense. CONGRUENCES WHOSE MODULI ARE IDEALS. 397 since a f = I, mod p. The integers n) are, moreover, incongruent each to each, mod p (Th. 29), and being t in number, are, therefore, all the roots of 12), since 12) cannot have more than t incongruent roots (Th. 25, Cor. 2). But every integer appertaining to t must evidently be a root of 12) and we need look, therefore, only among the integers 1 1 ) to find all the integers belonging to t . Let a r be as before any one of the integers 11). If a r appertain to t we must have a r ,a 2r , •••,a u_1)r all incon- gruent to 1, mod p. By Th. 30 the necessary and sufficient condition for this is ir^o, mod t, 13) where i runs through the values 1, 2, ••••, t — 1. It is easily seen that the necessary and sufficient condition that 13) shall hold is that r shall be prime to t. Hence the necessary and sufficient condition that any one a r of the integers 11) shall appertain to t is that its exponent r shall be prime to t. This condition is fulfilled by <f>(t) of the integers 11), and we have proved therefore that ^(f) =either$(£) oro. We shall now prove that the latter case can never occur. We separate the <f>(p) integers of a reduced residue system, mod p, into classes according to the divisor of <j>(p) to which they appertain; that is, if t 1 ,t 2 , --,t n be the positive divisors of <f>(p) we put in one class the \p{t x ) integers of the above system that appertain to t x , in another class the if/(t 2 ) integers that apper- tain to t 2 , etc. It is evident that no integer can belong to two different classes and that every integer of this system must belong to some one of these classes. The integers of a reduced residue system, mod p, being <£(p) in number, we have, therefore rt*i)+rth) + "'++(**) =+(p>* But, considering <£(p) simply as an integer of R, we have also (Chap. Ill, Th. 6) *('i) +<K' 2 ) + — +*(*•) =*(P). 398 CONGRUENCES WHOSE MODULI ARE IDEALS. Hence iK'i) +<H' 2 ) H hiKf») =*d) + <K' 2 ) H h *('»). 14) Since, however, every term in the first member of 14) is equal either to the corresponding term in the second member or to o, and hence, if even a single term in the first member of 14) were o, 14) would not hold, no term in the first member of 14) is o. Therefore f(t) =<f>(t). An examination of the table (p. 393) will illustrate this. §27. Primitive Numbers with respect to a Prime Ideal Modulus. 1 Among the integers of a reduced remainder system, mod p, there are, we have seen, <£(<£(£)) that belong to the exponent <f>(p). These integers are caller primitive numbers with respect to the modulus p, or briefly, primitive numbers, mod p. From the table (p. 393) we see that 3 and 5 are primitive num- bers with respect to the modulus (7, 3 + V — 5)- If p be a primi- tive number, mod p, the <f>(p) powers of p, n°= I n 1 n 2 n 3 ••• n^^)- 1 P x > P } P > R > > P y form a reduced residue system, mod p. This is for many pur- poses an extremely useful way of representing such a system. We can determine a primitive number, mod p, by the method used (Chap. Ill, § 33) to determine a primitive root of a rational prime. We can prove Wilson's Theorem for an ideal modulus by the aid of such a reduced residue system, just as the original theorem was proved for rational integers (Chap. Ill, § 29). It will be noticed that the primitive numbers, mod p, play exactly the same role with regard to p that the primitive roots of a rational prime, p, do with regard to p. It would seem desirable to have the nomenclatures the same, but those employed are the usual ones. It would, perhaps, be best to use the term primitive number instead of primitive root in the case of rational integers. § 28. Indices. // CL = p l , mod p, 'See Hilbert: Bericht, §9. CONGRUENCES WHOSE MODULI ARE IDEALS. 399 where p is a primitive number, mod p, and i be one of the num- bers o, I, 2, .--, <p(p) — i, i is said to be the index of a to the base p with respect to the modulus p. The relation between an integer and its index, which was seen in R to be similar to that of a number to its logarithm, is evidently the same in the case of ideals. It can be shown exactly as in R that, if p be any primitive number, mod p, a, (3 any integers of the realm, and m a positive rational integer, we have the follow- ing relations. i. The index of the product of two integers is congruent to the sum of the indices of the factors, mod <j>(p), that is; ind p (aft) = ind p a + ind p /?, mod <f>(p) . ii. The index of the mth power of an integer is congruent to m times the index of the integer, mod <f>(p), that is; ind p a m = m ind p a, mod cp(p). We observe that in every system ind p I = o. By means of the following tables we can illustrate the use of indices for an ideal modulus. Table A gives for the modulus (7> 3 +V — 5) the index to the base 3 of each integer of a reduced residue system, and Table B gives the residue corre- sponding to any index to the same base and modulus. It is evident that two integers congruent to each other, mod p, have the same index in any system of indices, mod p. A. Residue 1 2 3 4 5 6 Index 2 1 4 5 3 B. Index 1 2 3 4 5 Residue » 3 2 6 4 5 400 CONGRUENCES WHOSE MODULI ARE IDEALS. To pass from an index system with the base p x to one with the base p 2 , the modulus being p, we find as in R that ind p2 a = ind pi a • ind^ p v mod <p(p) ; that is, to obtain the system with base p 2 from one with base p lt we multiply each index of the latter system by ind p p lf the smallest positive residues, mod <p(p), of these products bring the required system to the base p 2 . In particular, if a = p 2 , we have ind^/yind^EEE i, mod </>(». Ex. To obtain for the modulus (7, 3 + V — 5) a system of indices to the base 5 from one of the base 3 we have first to find ind 5 3. From the relation just given ind 3 5 • ind., 3^ 1, mod 6, whence from Table A it follows that 5 ind 5 3^1, mod 6, and therefore indi 3 = 5- Multiplying by 5 each index to the base 2 and taking the least posi- tive residues, mod 6, of these products, we obtain for the modulus (7, 3 + V — 5) the following table of indices to the base 5. Residue | 1 2 3 4 5 6 Index 4 5 2 1 3 § 29. Solution of Congruences by Means of Indices. As in R, the solution of any congruence of the form ax = /3, mod p, 1) where a is not divisible by p, can be effected by means of a table of indices for the modulus p ; for from 1 ) it follows that ind a + hid •*" = ind (3, mod 4>(p), which gives ind x= ind (3 — ind a, mod <f>(p), from which x can be determined. CONGRUENCES WHOSE MODULI ARE IDEALS. 4OI Ex. i. From the congruence (2 + V— ^arsss— I+SV^, mod (7, 3 + V^), we obtain ind 3 (2 + V — 5) -}- ind 3 ^^ind 3 ( — I+3V — 5), mod 6; that is, since 2 + V" 3r 5 = 6, mod (7, 3 + V =r 5), and — I^V^ss* mod (7, 3 + V^ r 5), 3 + ind 3 ;r ^ 4, mod 6, or inda X = I, whence *==3, mod (7, 3+ V - 5)- The solution of the congruence cur* mm f}, mod p, 2) where a is not divisible by p, can be reduced by the use of indices to the solution of a congruence of the first degree, mod <j>(p). From 2) it follows that inda-j- winder = ind/?, mod cf>(p), and hence n'mdx = md/3 — ind a, mod <j>(p), 3) which is a congruence of the first degree in the unknown x. Moreover, n, ind x, ind /?, ind a and <f>(p) are evidently to be regarded merely as integers of R. Hence by § 14 the necessary and sufficient condition that 3) shall be solvable, is that ind ft — ind a shall be divisible by the greatest common divisor, d, of n and <f>(p), an d, if this condition be satisfied, 3) has \d\ roots. To these \d\ values of ind x correspond \d\ values of x satis- fying 2) and incongruent, mod p. These are the roots of 2). We see therefore that by the use of a table of indices we can reduce the solution of both 1) and 2) to the solution of con- gruences between rational integers. Ex. 2. Consider the congruence (i + V^)* 4 ^— V^5, mod (7, 3 + V zr 5), 4) where 1 + V — 5 is not divisible by (7, 3 + V — 5) • ind 3 (1 + V — 5) +4 ind 3 x^= ind 3 — V — 5, mod 6; that is, since i-hV^SssS. mod (7, 3 + \A =r 5), 26 4-02 CONGRUENCES WHOSE MODULI ARE IDEALS. and — V— $ss3. mod (7, 3 + ^/^5), using table A, 5 + 4 ind 3 x ^= 1, mod 6 or 4 ind 3 x^2, mod 6. 5) Since the greatest common divisor, 2, of 6 and 4 divides 2, the con- gruence 5) has two roots which are easily found to be 2 and 5. Hence we have ind 3 j = 2 or 5, and therefore x==2 or 5, mod (7, 3 + V — 5). These results are easily verified by substitution in 4). We obtain (1 + V^)2 4 = 2 + 2V^ r 5== — V^5, mod (7, 3 + V^5), and (i + V^5)5* = 2 + 2V Tir 5 = — V^5, mod (7, a + V^ 1 !)- Ex. 3. The congruence (1 + V z=r 5)^^2, mod (7, 3 + V^ r 5), has no roots, since the congruence ind 3 (1 + V — 5) +4 md 3 * = ind 3 2, mod 6, or 4 ind 3 x s 3, mod 6, has no roots, the greatest common divisor, 2, of 4 and 6 not dividing 3. Ex. 4. Construct a table of indices to the base 10 for the modulus (23, 8 -f- V — 5) and solve by its aid the congruence (2-\-3yy-~5)^ = -^/^5, mod (23, S + V" 11 !). Ex. 5. Show that the congruence (i + V=S)**e=i5, mod (23, 8 + V^5) has no root. The congruence x n s j3, mod p, where p is a prime ideal, can be treated as was the corresponding congruence in R (Chap. Ill, § 34), and a criterion for its solvability given analogous to Euler's. The general congruence of the 2d degree in one unknown can be discussed and the first part of the theory of quadratic residues for ideal moduli developed as in R, Legendre's symbol being replaced by ( - J , where a is an integer and p a prime ideal of k(Vm) (see Sommer: Vorlesungen iiber Zahlentheorie, pp. 92-98). The reader should work out the above. It is evident from the nature of an ideal that no direct reciprocal relation can exist between a and p, such as that between two rational primes as expressed bv the quadratic reciprocity law. A discussion of the reciprocity laws in the higher realms is beyond the scope of this book; for them the reader may consult Hilbert : Bericht, and Math. Ann., Vol. 51; Sommer: V. u. Z., Fiinfter Abschnitt. CHAPTER XIII. The Units of the General Quadratic Realm. § i. Definition. The units of any quadratic realm are those integers of the realm which divide every integer of the realm. For purposes of investigation they may be defined as follows : i. The divisors of I and hence those integers whose recip- rocals are integers. ii. Those integers whose norms are ± I. These two definitions are easily seen to coincide ; for, if c be a unit of &(Vw), we have from i ea=i, i) where a is any integer of k(\/m). From i) it follows that «[e]«[a] = I, and hence n[e] = ± I ; that is, ii is a consequence of i. Likewise, if e be a unit of &(\/m), we have from ii ee'=±I, where e is the conjugate of c and therefore an integer of k(^m). Therefore e is a divisor of I, and hence i is a consequence of ii. It follows from the above definition that if each of two integers, a, /?, divide the other, their quotient is a unit; for, if a//? = y, y and i/y are both integers ; hence y is a unit by i. In particular, the quotient of two units is a unit. In investigating the units of the general quadratic realm, we shall distinguish two cases accord- ing as the realm is imaginary or real. 403 4O4 THE UNITS OF THE GENERAL QUADRATIC REALM. § 2. Units of an Imaginary Quadratic Realm. The fact that the norms of all the integers of an imaginary quadratic realm are positive will enable us to determine the units of such a realm. Let m be a positive integer containing no squared factor; then &(V — m) is an imaginary quadratic realm, and we have seen that all imaginary quadratic realms will be obtained if m take all positive values. Let e, = x -\-yio, be a unit of k(\/ — m), 1, w being a basis of the realm. We have n[ € ] = (x-\-yo>)(x + y»') = i, 1) the value — 1 being impossible, since the realm is imaginary. We have now to see for what rational integral values of x and ti) holds, and to do so must distinguish two cases. i. When — m = 2 or 3, mod 4, and hence w=y — m. Then n [e] = (x -L- y V — m) (x — y V — m) = x 2 -\- my 2 = 1. If m > 1, it follows that y = o and x=± 1, and hence c = ± 1. If m=i, we have the realm k(i) whose units we have found to be ± 1, ± i. ii. When — fnasi, mod 4, and hence <a= (1 +V — m)/2. Then 1 -f V — m\f 1 — y / —m\ r ., / 1 + V —m\( 2 my 1 + — -* I- 4 If m > 4, it follows that y = o and a-=± i, and hence c=± 1. If w = 3 we have the realm fc(V — 3) whose units we have found to be ± 1, =b[(i ±V — 3)/ 2 ]- We see, therefore, that k(i) has the four units ± /, ±i, and k(y — 3) the six units ± 1, db [(1 ±V — 3)/ 2 ]> an d that all other imaginary quadratic realms have only the two units ± /. THE UNITS OF THE GENERAL QUADRATIC REALM. 405 § 3. Units of a Real Quadratic Realm. The determination of the units of a real quadratic realm is much more difficult. We shall see that, as in the realm k(^/2), the units of such a realm are infinite in number and can all be expressed as powers of a single unit called the fundamental unit. To show this we shall need the two following theorems, the first of which, due to Minkowski, is of great importance in the theory of numbers. Theorem i. // cl x x -\- p x y, a 2 x-\-/3 2 y be two homogeneous linear forms with real coefficients whose determinant CL ft 8 = is not 0, there exist tzvo rational integers, x , y , not both zero such that and If we put then |<*i*o + A:yo|i|V8|, 1*2*0 + &yolslV*|. 1 x=^t A. A 1) 7, or Putting y^A.i + B.r, B. 2) we see that A8= 1. If now we can find two quantities, £ , r] , such that I&J^I/IVA] and ho|^i/|VA[, 1 Minkowski: Geometrie der Zahlen, p. 104. Hilbert: Bericht, Hulfsatz 0. 406 THE UNITS OF THE GENERAL QUADRATIC REALM. and such that the corresponding values x , y of x and y are rational integers, then x and y are the required values of x and y. For, if ^'o^^i^o + ^o. and 3'0 = ^2& + -^2'70» then rj = a 2 x + p 2 y , and hence, since |«o|^|V«l and h |= we have l^-o + tool^lVSl. To prove our theorem it will be sufficient therefore to show that two quantities, | , r} , exist which satisfy the conditions |€o| ^i/IVA| ; hfo|si/|VS|, and such that are rational integers, where A lt A 2 , B x , B 2 are real and A = A B, A b 2 4=0. In the proof of the theorem we shall prove first the case in which a lt a 2 , fi lf (3 2 are rational and integral, then that in which the coefficients are rational and finally require merely that they be real. In the first two cases the theorem will be proved in its original form, in the last case in the equivalent form given above. The proof in the second case will depend directly upon the truth of the theorem for the first case, and that in the third case upon case two. i. Let a lf a 2 , f} lf fi 2 be rational integers. We shall need a theorem concerning binary linear forms. Calling a binary linear form a x x + b x y, where a lf b x are ra- THE UNITS OF THE GENERAL QUADRATIC REALM. 407 tional integers, for the sake of brevity a form, and two such forms a form system, we say that a form c x x -f- d x y is reducible to o by the form system a x x + b x y, a 2 x -f- b 2 y, if c x x + d x y = g x (a x x + b x y)+g 2 (a 2 x + b 2 y), where g x , g 2 are rational integers. Two forms are reducible to one another by a given form system if their difference is reducible to o by this system. Two form systems are said to be equivalent if every form that is reducible to o by either one of the systems is also reducible to o by the other system. The analogy to the basis of an ideal is at once evident, for, if a x s=s (a x u> x + b x w 2 , a 2 (o x -f- b 2 oy 2 ) be an ideal, where a x <a x -j- b 2 o) 2 , a 2 o) x -f- b 2 o) 2 is a basis, then an integer, c x (o x -)- d x a> 2 , is a number of the ideal if c x (o x + d x <o 2 = g x (a x (o x + b x (o 2 ) + <7 2 ( a 2 w i + o 2 <o 2 ), where g x , g 2 are rational integers. Thus the reducibility of a form to o by a given form system corresponds to a number be- longing to an ideal. We can show exactly as in the case of a canonical basis of an ideal (Chap. XI, Th. i) that among the form systems equivalent to a given system there is one, Ax, Bx -f- Cy, such that among all forms of the form ax, reducible to o by the given system, Ax is that one in which a is smallest in absolute value, and among those of the form bx -f- cy reducible to o by the given system, Bx -\- Cy, is one of these in which c is smallest in absolute value. We can then show that, if two form systems be equivalent, the absolute values of the determinants of their coefficients are equal (see Chap. XI, Th. i, Cor.). It will now be evident that to say in the case of forms that two forms are reducible to one another by a given form system is the same as saying in the case of an ideal that two integers are congruent with respect to this ideal, for in the former case the difference of the two forms is reducible to o by the given system while in the latter the difference of the two integers is a number of the ideal. 4o8 THE UNITS OF THE GENERAL QUADRATIC REALM. The statement in the one case that there are exactly a i K a 2 K forms, no two of which are reducible to one another by the form system a x x + b x y, a 2 x -\- b 2 y, is the same as the statement in the other case that there are exactly a 2 b 2 integers which are incongruent each to each with respect to the ideal {a x u x + a 2 w 2 , b x w x -f- b 2 o> 2 ), and may be proved similarly (see Chap. XII, Th. i). We observe now that |8| is equal to one of the square numbers i, 4, 9, 1 6, 25, -.., r 2 , (r+1) 2 , or lies between two of them. Let r*g|*f< ('+l) a . and form the (r+ i) 2 forms ra = o,i,2,...,r, ax + by A 1 J \ b = o, 1,2, ->,r. 3) Since there are only \B\ forms, no two of which are reducible to one another by means of the form system a x x -\- (3 x y, a 2 x -f- (3 2 y, at least two of the forms 3) are reducible to one another by this system. Let these two forms be a\x -\- biy and tyx -\- bjy. Then dix + b t y = ajx + b $ y + c(a x x + (3 x y) + d(a 2 x + fty ) ; that is, (a t — a,)* + (bi — bj)y= (a x c + a 2 d)x + (ftc + ftd)y, and hence o^c + a 2 d = a< — a,-, THE UNITS OF THE GENERAL QUADRATIC REALM. 409 Since \ai — aj\ and |&« — bj\ -^r, they are both g | \/8\ ; hence c and d are the required values of x and y. ii. Let a t , a 2 , ft, ft be rational fractions. Let their least common denominator be g. Then #a x , #a 2 , #ft, </ft are rational integers. By case i we can find two rational integers, x , y , such that \g<*z*o + #ft3'o|^ I V¥ 2 |- 4) On dividing both members of 4) by g we get k* + ft3'o|^|V8|, K*o + ft3'o|^|V8|. Hence x and y are the required values of x and y. iii. Let a lt a 2 , ft, ft be any real numbers. We shall prove the theorem in its second form ; that is, that if A lf A 2 , B lt B 2 be any real numbers, such that the determinant, A b 2 \ is not zero, there exist two numbers, £ , rj , satisfying the conditions |&|*X/|V*|. ho|^l/|VA|, and such that x = A 1 £ -f- B x r) , y =A 2 $ -\-B 2 rj , are rational integers. Let A lf A 2 , B lf B 2 be defined respectively by the rational fun- damental series a lt a 2 , a z , •• b lt b 2 , b 3 , ■■ Ox's <*2> a %> ■ • 5) that is, ^ 1 = lim a n , B 1 =lhn b, A t> = \im (in, J5„ = lim bj 6) 410 THE UNITS OF THE GENERAL QUADRATIC REALM. Let A = a b' where a n , b n , a n ', b n ' are the nth terms of the above series, then lim A n = lim a n • lim b n ' — Hm a n r • Hm b n , = A 1 B 2 — A 2 B 1 = &. We observe now that in the series * A x , A 2 , A 3 , -.., 7) though some of the terms may be o, the number of such terms is always finite ; that is, from some ith term onward no A is o ; for otherwise, lim A n would not exist or else would be o. Since now the terms 5) are all rational numbers, and A* and all succeeding A's are different from o, we can find by case ii for every set, a i+p , bi +p , a' i+p , b'i +p of (i-{-p)th terms of the series 5), two numbers, & +p , rji +p , such that |&*| s-i/! V^Ji#|.. h* + p|ii/|V^|, , 8) and that 0* +p&+p + bi +v -qi +p , a'i+p£i+ P + b f i +P Y)i +Pf are rational numbers. From 8) it is evident that the terms of the series rji, rji +1 , rji +2 , •••, have an upper limit, for no term of the series |Ai|, |A {+1 |, |A 4+2 |, -.., is o, and lim A n = A =4= o, whence the terms of this series have a lower limit. Let this upper limit of the £'s and */s be k. Consider a system of rectangular axes and construct a square THE UNITS OF THE GENERAL QUADRATIC REALM, 41 with the origin as center, its sides equal to 2k and parallel to the axes. v If now we consider £i +p , rji +p as the abscissa and ordinate re- spectively of a point, we may represent each pair of numbers £up, yi + p(P = o, 1, 2, • • • ) by a point. All these points will be within or on the boundary drawn as above. Since there are infinitely many points (& +p , r}% +v ) within or on this boundary they will have at least one limiting point within or on the boundary. Let the coordinates of this point (or, if there be more than one, of any particular one) be £ , rj . There will be certain series of the points (|i +p , r)i +p ) which approach and remain arbitrarily close to (£ , rj ) as p is indefi- nitely increased. If (f^y, y i+ /) denote such a series, where p' represents only those values of p which gives this series, we have £0 = lim £*+,'» % = lim W Then P'=o lim ("i+Jiw + **f«*w) = A S + B %> p'=x> •im «,yf„y + b\ +p ,r,. +p ) = A% + B' % . P '=«3 But all terms of the series and are rational integers. a 'i+A+p' + ^i+pViW 412 THE UNITS OF THE GENERAL QUADRATIC REALM. Hence their limits, A$ -f- B-q Q and A'£ -f- B'-q 0i are rational in- tegers. Therefore £ and r/ are the required numbers, and the theorem is proved in its second form. It holds therefore in its original form. From the above theorem we have at once the following theorem : Theorem 2. If a x x + fS x y, a 2 x -f- (3 2 y be two homogeneous linear forms with real coefficients, whose determinant «i Pi "1 ft is not and k, k k be any two positive quantities such that K*k= | 8 I> there exist rational integers x , y , not both 0, such that Given the two forms L ft a 2 , ft 1. whose determinant is not zero, there exist by Th. 1 two rational integers, -x , y oy not both o, such that |*i , ft u \fC ° K J °\ ' «. _L ft Z^ + IT* ^ 1 'A ,V A and hence l^i^'o + Ay | g *, |*t*t + toils «* From this theorem we obtain at once the following theorem, which is necessary for the investigation concerning the units of a real quadratic realm as well as interesting on its own account. 1 Hilbert : Bericht, Hulf satz 7. = a/ — tu = Vd 4= O, THE UNITS OF THE GENERAL QUADRATIC REALM. 413 Theorem 3. There are in every real quadratic realm an in- finite number of integers, the absolute value of whose norms i. The existence of at least one such integer is seen at once. For, if 1, to be a basis of the realm, x -f- yoj, x + y*' are two linear forms whose determinant 1 &) 1 *»'! and making use of Th. 2 and putting K=K X , K^ = \\/'d\/K 1 , where * t > o, we see that there exist two rational integers, x x , y lf which are not both o, and which are such that k + ^'I^IWlAi, and hence I {*i + &•) Oi + 3'X) f s I V^| ; that is, M*i+*rf|*'|V2|r Therefore the realm contains at least one integer, a 1 , = .r 1 + >'i w > the absolute value of whose norm is less than or equal to |V^|- To show that there are an infinite number of such integers we proceed as follows : To prove the existence in the realm of an integer, a 2 , = x 2 -\- y 2 o>, that is different from ± a lt and such that \n[a 2 ]\^\Vd\, we have only so to choose k, that from the condition it will necessarily follow that a 2 ^±a v 4 14 THE UNITS OF THE GENERAL QUADRATIC REALM. This may be effected in infinitely many ways, a simple one being to take for k ± some positive quantity /c 2 <|a 1 |; for example, \aj2\, for then from \^2[^ K 2< K x it follows that |a,| < \a x \, whence a 2 =j= ± ol ± . Since by Th. 2 there exist two rational integers, x 2 , y 2 , which are not both o and which are such that \x 2 + y 2 «> |^K 2 , it follows that there is in the realm an integer, a 2 , = x 2 -f- y 2 w, different from dfc a ly and such that |»la,]|s|V3|. To prove the existence in the realm of a third integer, a 3 , dif- ferent from ± a.! and ± a 2 and such that we have only to put for k in the inequality a positive quantity k 3 less than k 2 , when it is at once evident that such an integer exists; for from \a 3 \ g* 3 <[a 2 |<I a il it follows that a 3 =f= ± a 2 , and a 3 4= ± a x . Continuing in this manner we can prove the existence in the realm of as many such integers as we choose. They are, there- fore, infinite in number. Ex. We shall illustrate the above theorem by showing that we can actually find in k{yjy) as many integers as we please, the absolute values of whose norms are less than or equal to | V 2 ^ I, d being in &(V7) equal to 28. Following the method employed in the proof, we let a.i, = Xi -\- yiV7> be any integer satisfying the required condition and * be any positive quantity, say 2. We have to determine x, y so that \a,\i^Bi, W\f 9) THE UNITS OF THE GENERAL QUADRATIC REALM. 415 We may assume without loss of generality that Xi, yi have the same sign, for, this assumption being made and V7 being taken positive, I X! + yiV7 1 > I xi — W7 I, otherwise not, and the most favorable way in which the conditions 9) can be imposed is | x x + .ViV7 I = tne larger of the two quantities k and I V d I A, here | V 2 & |/2, | Xi — yi\/7 | ^the smaller of the two quantities k and I V^ |/ K > here 2. Making this assumption, the conditions which Xi, yi must satisfy are |*t-f W7|iV7, 10) \x 1 + y 1 y/7\^2. 11) The further assumption Xi, yi positive, which may evidently be made without loss of generality, will simplify the work. Doing this, we see that, since Xi and yi have the same sign, *i = 0, yi = i Xi=z I, yi = o Xi = 2, yi = o. or or But it is evident from 11) that of the three values only those pairs in which yi = are admissible ; hence xi + yiV7 = I or 2. The only integers of k{\/j) which satisfy the condition 9) are therefore ± 1, ±2. The absolute values of the norms of 1, — 1, 2, and — 2 are evidently all less than | V28 |. To find another integer a«, = x 2 + y 2 \/7, the absolute value of whose norm is less than | V 2 8 |» we proceed as in the proof of Th. 3 and let k= I a,/2 I, where a x is any one of the integers 1, — 1, 2 or — 2, say 2; that is, we have now to determine x 2 , y% so that I x 2 + y 2 V7 1 ^ V28. 12) 1*2— y 2 V7l^i, 13) where Xz, y 2 are assumed to be both positive. Since x 2 , y 2 have the same sign and the value o for a 2 is excluded, we see from 12) that x 2 = 0, y 2 = 1 or 2, or #2=1 or 2, y 2 = 1, or x 2 s= 1, 2, 3, 4, 5, y 2 = ; but 13) excludes all these values except x 2 = 2, y 2 — 1 and x 2 = 1, y 2 = o. The last set gives a 2 = 1, an integer already found, but the other gives cc 2 = 2-j- yjy, a new integer satisfying the conditions 12) and 13), and hence one the absolute value of whose norm is less than | V 2 ^ |. 4l6 THE UNITS OF THE GENERAL QUADRATIC REALM. We see indeed that l»[2 + V7]|=3<l V&l If now we put *=s| (2 — V7)/ 2 l> and proceed as before, we can find an integer a 3 such 4hat I n [a 3 ] I < I V28 I, and a, =|= ± a* a, =)= ± a* Continuing in this manner, we can find as many integers as we please satisfying the required conditions. Theorem 4. // * be any positive constant, there exist only a finite number of algebraic integers of the second degree such that they and their conjugates are simultaneously less than k in abso- lute value. 1 Let a be an integer of the second degree such that |a| <*, |.a'| < K . 14) Let x 2 -\- a x x -j- a 2 = o be the irreducible rational equation of which a and a' are the roots. We have a 1 = -*-(a-\-a'), a 2 = aa', hence |a x |=|a + a'|, |a2|=| aa '|- But \a-\-a'\<2 K , \aa'\<K 2 , hence \a x \<2 K , |o 2 |<k 2 . 15) It is evident that only a finite number of rational integers can satisfy the condition 15) ; hence there are only a finite number of equations of the second degree whose roots satisfy 14). There are, therefore, only a finite number of integers of the second degree satisfying 14). This theorem, it will be observed, is proved not for a single quadratic realm but for the integers of all quadratic realms taken together. Moreover, it will be noticed that not all the roots of these equa- tions satisfy 14) but that among their roots are all the integers of the second degree that satisfy 14). See Ex. § 4. 1 Hilbert : Bericht, Satz 43. THE UNITS OF THE GENERAL QUADRATIC REALM. 417 Theorem 5. There exists in every real quadratic realm a unit, e, different from ± 1, and such that every unit, 77, of the realm has the form rj=±e m , where m is a positive or negative rational integer, or o. 1 The proof of this theorem may be conveniently divided into the following four parts : i. Every real quadratic realm contains an infinite number of integers, a lf a 2 , a s , •••, the absolute values of whose norms are less than or equal to \^Jd\. ii. A quadratic realm, whether real or imaginary, contains only a finite number of ideals whose norms are less than \Vd\, and hence the infinite series of integers, a u a 2 , a 3 , ••-, considered as principal ideals, (a x ), (a 2 )> ( a s)> '"» 9^ ve on h a finite number of different principal ideals, whence it follows that the integers, a lf a 2 , a 3 , •••, must fall into a finite number of classes, each con- taining an infinite number of integers which differ from each other only by unit factors, and hence there are in every real quad- ratic realm an infinite number of units different from ± 1. iii. Infinitely many of these units of a real quadratic realm are greater than 1 ; among these there is a smallest one, c. iv. Every unit, rj, of the realm has the form where m is a positive or negative integer, or 0. Having already proved i, we begin with ii. ii. We obtain all prime ideals whose norms are less than | \/d\ l)y resolving all positive rational primes less than | \/d\ into their prime ideal factors. There are evidently only a finite number of such prime ideals. By multiplying these prime ideals together we obtain all ideals whose norms are less than |V^|- These ideals are evidently also finite in number. Hence among the infinite system of principal ideals (<*i)»(a2)»(a«)» •'•> 16) a See Hilbert: Bericht, Satz 47. 27 41 8 THE UNITS OF THE GENERAL QUADRATIC REALM. whose norms g| V^|> at least one ideal must be repeated an infinite number of times. Let the infinitely many ideals (3*1)1 (a* 2 )> (g# 8 )* taken from the system 16) be the same. Then each one of the integers a iv CLi 2 , at 3 , •••, 17) must be divisible by every other one ; that is, we have a il = /3a i2 , and ya.i 1 = a.i 2 , where (3 and y are integers. Hence /3 and y are units (§ 1), and are, moreover, different from ± 1, since we may assume that no two of the integers 17), as a*!, cii 2 , are so related that 3^=4= ± CLi 2 . Furthermore, the number of such units is infinite ; for ai 1 ==Sai 3f where 8 is a unit, and if ft =±8, then a i2 =^±a i . i , which is impossible. Hence the quotients obtained by dividing each of the integers 17) by a« x constitute an infinite system of units, Vi, %j ■••> such that we never have r)i = ± rfj. iii. There are in the realm an infinite number of units which are > 1 ; for from each one of the units, rf lt rj 2 , • • •, as rji, we can derive such a unit, since one of the integers, all of which are units, must be such a unit. Among this infinite system of units greater than 1 there is a smallest one; for, if rji THE UNITS OF THE GENERAL QUADRATIC REALM. 419 be a unit greater than i, there are by Th. 4 only a finite number of integers, a, of the realm such that |a|<i^; |a'|<T7i; and hence only a finite number of units, rj, such that \y\<ii> W\<vi- l8 ) But if 7] be any unit greater than 1 but less than rji, we have from W=± 1, r?b'|= I, • and hence |?/|<i< 77*; that is, r] must satisfy 18). There are, therefore, only a finite number of units, rj, such that and hence there is among them a smallest one, which is, of course, the smallest of all those units of the realm that are greater than 1. Denote this unit by e. iv. It is evident that the units •••, ±e" 2 , ±e-\ ±€°, zte 1 , ±€ 2 , -.., 19) are all different; for from e m =±e n , m > n } it would follow that e m- n==z± . If which is impossible, since e=4=± 1, and none of the numbers of the realm are imaginary. We shall now show that the system 19) comprises all units of the realm. Let £ be any positive unit greater than or less than 1 ; then £ will lie between two consecutive, positive or negative powers of c, or else be equal to a power of e; that is, we can determine an integer, n, positive, or negative, such that € M <£<e n+l 420 THE UNITS OF THE GENERAL QUADRATIC REALM. Let !/€»=&; then |i is a unit, and we have i si li < «• But we cannot have K & < €, for 6 is the smallest unit greater than i. Hence and therefore £ = e n . When n is positive the units are greater than i, and when n is negative they are all positive but less than i; n = o gives £=i r By letting n take all rational integers from — cc to -{- cc we thus obtain all positive units of the realm. Now let £ be a negative unit; then — £ is a positive unit, and we have — $=e n ; hence $ = — e n . Every unit, $, therefore, of a real quadratic realm has the form where n is a positive or negative rational integer, or o, and c is the smallest unit of the realm > 1. This unit e is called the fundamental unit of the realm. § 4. Determination of the Fundamental Unit. If in any quadratic realm k(\/m) any unit, rj, be known, we can at once obtain a unit greater than 1 ; for one of the four units, rj, —7), i/rj or — i/ v , has this property. Denote that one of these four units which is greater than 1 by 7) 1 . We have now to determine whether there are any units in the realm which are greater than 1 but less than rj u and, if there be any such units, to find the smallest of them. THE UNITS OF THE GENERAL QUADRATIC REALM. 42 1 Th. 4 enables us to do this; for by the method employed in the proof we can find the rational integral equations finite in number, among whose roots are the integers a of the second degree finite in number, such that \a\<*; \a'\<m> i) Among these integers will be included all units, |, such that K ! < %, 2) for we have seen that from 2) and ££'=±h it follows that |f|<*; |TK*. Since we wish to find only those units which satisfy 1), and the last term of the irreducible rational equation satisfied by an integer of the second degree is the norm of the integer, we may make the last term of each of our equations =t I. Writing down, therefore, all irreducible equations of the form x 2 + ax± 1=0, 3) where a is a rational integer, such that \a\<2 Vl , and solving these equations, we obtain all units which satisfy 1), not only of the realm under discussion but of all real quadratic realms. If there be any unit of the realm under discussion which is greater than 1 but less than rj 1} it will be found among these. Ex. Let the realm under discussion be £(V5)- Since «[2 + V5~]= — 1 2 + V5^is a unit of £(V5)- Moreover 2-}-\/5>i- To determine those units of &(V5) that are greater than 1 but less than 2 -f- y/s, if any exist, we write down all irreducible equations of the form 3), in which | a | < 2(2 -|- V5)- We need only write those in which a is negative since the change of sign of a merely changes the signs of the roots. 422 THE UNITS OF THE GENERAL QUADRATIC REALM. We have, therefore, as the equations among whose roots will be found the unit sought for, if it exist, x- — X +1 = x 2 — x — J [ =0 X 2 — 2X + I = X 2 — 2X — ] [ =0 x 2 — 3x + I = o x 2 — 2,x—] [ =0 x 2 — 4x + i = o x 2 — 4x — ] [ =0 x 2 — 5* + i = o x 2 — $x — 3 [ =0 x 2 — 6x + i = o x * — 6x— J [ =0 x 2 — 7x + i = o x 2 — yx — [ =0 x 2 — 8x + i = o x 2 — Sx — [ = Solving these equations, we obtain four units of &(V5) which are greater than i, £(i + V5XK3 + V5), 2 + V5 and K7 + 3V5), and of them evidently £(i + V5) 1S tne smallest and hence the fundamental unit. The foregoing determination of the fundamental unit of a real quadratic realm depended upon the supposition that some unit of the realm was known. To find some unit of the realm we may proceed as follows, the method being that used in Th. 5 to show the existence in such a realm of a unit different from ± 1. Let k(\Ztn) be the realm. Determine first how many different ideals have their norms less than |\/d|. This is easily done by factoring all rational primes less than |V^| an d forming all products of these ideals, such that the norms of these products are less than |yd|. Sup- pose that there are m different ideals whose norms are less than |yd|. Find now w+i integers whose norms are less than |Vd|, which can be done by the method used in the proof of Th. 3. The quotient of some pair of these integers whose norms have the same absolute value must be a unit. This method of determining the fundamental unit may be shortened by observing that, if c + d\/m be the fundamental unit of &(\/m), where c and d are either rational integers or rational fractions whose numerators are odd and denominators 2, then c and d are both positive, and hence no equation of the form 3), where |a|< 2c, can have as a root a unit of the realm greater than 1 and less than c + d^Jm. Therefore the funda- mental unit is a root of the first equation among the equations THE UNITS OF THE GENERAL QUADRATIC REALM. 423 3), arranged in ascending values of \a\, whose roots are units of k(\/m). From this, we see that, in the example above, it was unnecessary to proceed further after finding i(i+V5) as a root of x 2 — x — i = o. The number of equations to be examined may also be reduced by observing that we must have a 2 + 4 53 o, mod m, if an equation, whose last term is — i, is to have as a root a unit of k(y/m). If m be divisible by a prime, p, of the form 411 — I, this relation is evidently impossible, for it requires that — 1 shall be a quadratic residue of p. Hence the fundamental unit of k(\/m) can not have — 1 as norm, if m be divisible by a prime of the form 4n — /. § 5. Pell's Equation. It will be at once recognized that the determination of the units of a real quadratic realm, k(^/m), is equivalent to solving Pell's Equation : x 2 — my 2 =± 1, where 771 = 2 or 3, mod 4, and x 2 — my 2 = ± 4, or x 2 — my 2 =±i, where m==l, mod 4; furthermore the smallest solution will give the fundamental unit. The general problem of determining an integer with given norm, H, of which the above is a particular case, is evidently equivalent to solving x 2 — my 2 = H. The following theorems relating to Pell's Equation are taken from Chrystal's Algebra, Part II, p. 450, and the reader is referred to this work for their proof and for the complete discussion of this subject. 1 Confining ourselves now to solutions in which x and y are prime to each other, for, if x and y have a common factor r, then r 2 must be a factor of H and we can reduce the 1 See also H. J. S. Smith : pp. 192-200. 424 THE UNITS OF THE GENERAL QUADRATIC REALM. equation to x' 2 — my' 2 = H', where H'=H/r 2 , and limiting our discussion to the case |H|<| V*»|, we have the following theorem : Theorem 6. The equation x 2 — my 2 =s ±: H where m andH are positive integers and m is not a perfect square, admits of an infinite number of solutions provided its right-hand side occurs among the quantities ( — i) n M n belonging to the devel- opment of yjm as a simple continued fraction, zvhere M n is the (n-{-i) th rational divisor, and all these solutions are x = p n , y = q n , where p n /q n is the 11 th convergent in the development of V w. Cor i. The equation x 2 — my 2 =1 1 ) where m is positive and not a perfect square always admits of an infinite number of integral solutions, all of which are furnished by the penultimate convergents in the successive or alternate periods of yjm. Cor 2. The equation x 2 — my 2 = — 1 2) where m is positive and not a perfect square admits of an infinite number of integral solutions, provided there be an odd number of quotients in the period of "\/m, and all these solutions are fur- nished by the penultimate convergents in the alternate periods of \/m. If there be an even number of quotients in the period of V m the equation has no integral solution. If p, q be the first solution of 1) or 2) and we have x ~\-y V m=z (P ± o. V m ) n y where n takes all positive values, or all odd positive integral values. Then the resulting values of x, y are all solutions of 1) or 2) respectively. THE UNITS OF THE GENERAL QUADRATIC REALM. 425 For the discussion of the equation x 2 — my 2 ==±Hj where H is greater than \/m, the reader is referred to Chrystal's Algebra, Part II, p. 454. The following examples will illustrate these theorems : Ex. 1. Determine the fundamental unit of &(\/7)- We must solve x 2 — 7y 2 = — I, if possible, and if not possible, then x 2 — yy 2 =i. Expanding \/y in a continued fraction we have 1/7 = 2 + 1+ 1 + 1 1 ~+ 4 + + ill which gives the following table, where, as in Chrystal, n is number of convergent, an the wth partial quotient, p n the numerator of the nth convergent, q n the denominator of the nth convergent, M n the (n+i)th rational divisor. n * A In M n 1 2 2 I 3 2 1 3 I 2 3 1 5 2 3 4 1 8 3 1 5 4 37 14 3 There being an even number, 4, of quotients in the period of V7> the equation x 2 — 7^ 2 = — 1 has no solution (Th. 6, Cor. 2); that is, the realm k(yjy) has no unit with negative norm. We observe, however, that the penultimate convergent, 8/3, in the period of V7 gives 8 2 — 7 • 3" = 1, (Th. 6, Cor. 1.) thatjs, 8 + 3V7, 8 — 3\/7, —8 + z\/J and —8 — 3V7" are units of £(V7)> 8 + 3V7 being the fundamental unit. This can be verified by the method of the previous section. Ex. 2. Determine the fundamental unit of ^(V I 7)- Expanding yjiy in a continued fraction, we have > / ^ = t+8 i + 8T... which gives the table, there being only one quotient in the period of V J 7- n a n Pn In ^„ I 4 4 I 1 2 8 33 8 1 Hence the equations x 2 - 17 y 4 and x 2 — I7y" = 4 426 THE UNITS OF THE GENERAL QUADRATIC REALM. have no solution, but the equation x- — i/y' 2 = — 1 has the solutions x= ± 4, 3; = ± 1, and 4 + V*7 1S seen to be the funda- mental unit. This can be verified by seeing that among the roots of the equations x- — ax ± 1 = where \a\ < 2(4+ V T 7)> the only unit of k{\Jiy), which is greater than 1, is 4 + Vi7. _ Ex. 3. Find the fundamental units of the realms &("y/io), £(-^11) and CHAPTER XIV. The Ideal Classes of a Quadratic Realm. § i. Equivalence of Ideals. 1 We have seen (Chap. XI, Th. 5) that in any quadratic realm, ^(V«), there exists for every ideal a an ideal m, such that the product am is a principal ideal. Attention was also called to the evident fact that although the particular ideal which was shown to have the desired property was the conjugate a' of a, all ideals of the form ct'(y), where (y) is any principal ideal, have this property. Since, moreover, if a and 16 be any two ideals, there exists in a a number a such that (a) /a is prime to b (Chap. XI, Th. 11), it is evident that there is an infinite number of ideals each one prime to all the others and each such that its product by a is a principal ideal ; for, if a x be any number of 0, then (a 1 )=aa 1 , where a t is an ideal having the desired property. By the above theorem there exists in a a number a 2 such that (a 2 )~aa 2 , where a 2 is prime to a x and is evidently an ideal having the desired property. In like manner there exists in a a number a z such that (a 3 ) = aa 3 , where a 3 is an ideal having the desired property and prime to 0^2, and hence to each of them. Proceeding in this manner, it is evident that an infinite number of ideals exist each of which is prime to all of the others and such that, when multiplied by a, the product is a principal ideal. 1 Hilbert: Bericht, Cap. VII. 427 428 THE IDEAL CLASSES OF A QUADRATIC REALM. We see, therefore, that the ideal m need not contain a' as a factor; for example, (2, i+v=5)(3, i+V-5) = (i+V-5)> (2, i+y-5)(3, i_v-5) = (i-V— 5), where (3, i + V — 5) . an< 3 (3, 1 — V — 5) are prime to (2, 1 — V — 5) and to each other. From the fact that infinitely many ideals give, when multiplied by one and the same ideal, products which are principal ideals, we are led to the introduction of the idea of the equivalence of ideals, w T hich is defined as follows : Tzvo ideals, a and fc, are said to be equivalent if an ideal m exists such that the products am and bm are both principal ideals. The equivalence of a and B is expressed symbolically by writing a'r^ 6; that they are not equivalent by writing For example, as we have seen above, the product of each of the ideals (3, r~f-V— 5) anc * (3, 1— V— -5) by the ideal (2, 1 +V — 5) is a principal ideal; hence (3, 1 +V — 5) * s equiva- lent to (3, 1 — V — 5) j or m symbols (3, 1 +V— 5)^(3, 1— V— 5)- Likewise, since the product of (2, 1 +V — 5) by itself is a principal ideal, (2, 1 +V — 5) is equivalent to each of the two ideals (3, 1 +V ::r 5) and (3, 1— y/^). As an example from another realm k( V — 17), we see that (3, 1 + V— '17) ^ (ii, 4— V— ~I7), for it can be easily shown that (11, 4-fy— T7)(n> 4— V— ^7) = (ii)> and (11, 4+V— I7)(3> 1 +V— 17) = U+V— 17)> If a~b, then by the definition there exists an ideal c, such that QC=(ju), THE IDEAL CLASSES CF A QUADRATIC REALM. 429 Multiplying these equations respectively by b and a, we have abc=0)b=(»a; that is, if a^h, there exist two integers, /x and v, such that Furthermore, if a and b be any two ideals and there exist two integers, /x and r, such that O)a=0*)b, 1) then a ^ b ; for let m be any ideal such that the product am is a principal ideal (y), then multiplying i) by m, we have 0)am= (vy) = (/*)bm. But, if the product of a principal ideal and another ideal be a prin- cipal ideal, the second ideal must be a principal ideal also. Hence bm is a principal ideal and consequently ct^b. We may therefore define the equivalence of two ideals as fol- lows, this definition being, as shown above, exactly equivalent to the former one: Two ideals, a and b, are equivalent if two integers, a and (3, exist such that a(/B)==»(a).> For example, we have (i—V=~5)(3> i+V= 7 5) = (i+V ::r 5)(3 ) i— V=5), whence it follows that (3, i +V :::: 5) ~ (3> i — V^)- We shall use both of these definitions of equivalence, each having some advantages of its own. Equivalences between ideals obey the following laws: i. If a^b and b ^ c, 2) 1 Hilbert : Bericht, p. 223. 430 THE IDEAL CLASSES OF A QUADRATIC REALM. then a «— ' c, for from 2) it follows that there exist integers a, /?, y, 8, such that a(p)=Ha) and b(8) =c(y), and hence, multiplying these equations respectively by (8) and (a), a(/?S)=c(a y ), Therefore a /— ' C. ii. If a^b and c^b, 3) then qc r^ bb, for from 3) it follows that there exist integers a, /?, y, 8, such that a((3)=b(a) and c(8)=b(y), and hence ac(/?8) = bb(ay). Therefore ac ^ bb. If a^h, then from ii it follows immediately that where n is any positive rational integer. The original definition of equivalence given above is that used by Dedekind, the second is equivalent to the following, which is given by Hilbert and Weber: Every number of a realm, *, not an integer, can be represented as the quotient of two integers ; that is, If now we look upon a and /? as principal ideals and remove all factors common to (a) and (/?), we have (a)/G8) = a/6, a representation that is evidently unique. For example, let 1 + l/- 5 /€= 1-1/-5 THE IDEAL CLASSES OF A QUADRATIC REALM. 43 1 We have (1 + 1/^5) = (2, i + v^ Xz, 1 + v /: -j) = (3, 1 + v^) (1 - 1/- 5) (2, 1 + V- 5)(3, 1 - V- 5) (3, 1 - 1/--5) ' If inversely the quotient ct/b of two ideals, a and 6, where a and b may or may not have a common factor, is equal to the quotient of two principal ideals, (a) and (/?) ; that is, if a/b=(a)/(/?), and hence may be taken to represent in the above sense a number, k = ol//3, then we say that a is equivalent to b. For some purposes it is useful to define the equivalence of ideals in a narrower manner, considering a equivalent to b when and only when a number, k, whose norm is positive exists such that K = a/h; that is, when two integers, a and /?, whose norms have the same sign, exist such that (/J)a=(a)6. This definition of equivalence will evidently be essentially dif- ferent from the original one when and only when the realm con- tains no unit whose norm is negative. In quadratic realms this will always be the case except when the realm is real and the norm of the fundamental unit is — I. In general this definition of equivalence is identical with the original one in all realms of odd degree. Examples. Show that the following equivalences hold I) (23, 8— V— 5)^(7. 3+V— 5). 2) (7. i +V — 13) ~ (2. i+V— 13), 3) (, i+ f^y- ( .), 4) (2, V— 10) — (5, V— 10), 5) (3, 1 — V— 14) 2 — (2, V— 14), 6) (5, i+V26)-(2,V26), 432 THE IDEAL CLASSES OF A QUADRATIC REALM. § 2. Ideal Classes. Since, if two ideals, a 1} a 2 , be equivalent to an ideal a, they are equivalent to each other (§ i, i), the ideals of a realm can be separated into classes by putting two ideals into the same or different classes according as they are or are not equivalent to each other. The system of ideals composing such a class has the property that every ideal in it is equivalent to every other one and that it consists of the totality of all ideals which are equivalent to any one of the ideals composing the class. Such a class is called an ideal class and is denoted by a Latin capital letter. Any ideal a of a class A, may evidently be taken as the repre- sentative of the class, and the class is completely determined by a. The class composed of all principal ideals and as whose repre- sentative we can take (i), is called the principal class and is denoted by i. If a lt Q 2 be any two ideals of the class A, and b lf B 2 be any two ideals of the class B, then since from ctj. ^ o 2 , and &! r^ fc 2 , it follows that aj) x ^aj) 2 (§i,ii), it is evident that all ideals of the form ab, where a and b are any ideals of the classes A and B respectively, belong to a single class, C, which class can, however, contain infinitely many ideals other than the products ab. The ideal class C is called the product of the ideal classes A and B and we write C = AB. For example, we have (3, i + sF^S) (2, i + V^S) = (I + V~=~5), -whence it follows that the product of the classes of (3, 1 -f- V — 5) and (2, i + V — 5) is the principal class. But (3, 1 + V — 5) and <2, 1 + V — 5) belong to the same class, A. Hence we have A 2 = 1. THE IDEAL CLASSES OF A QUADRATIC REALM. 433 The product of any ideal class A by the principal class is A ; that is, A-i=A. Inversely from AB = B it follows evidently that A = r. In the multiplication of ideal classes it is evident from the definition of the product of two classes that the commutative and associative laws hold; that is, AB = BA and AB-C = A-BC. We see, therefore, that in the formation of the product of any number of classes, A 19 A 2 , •••, A m , the order in which the classes are taken will make no difference in the final result, which we denote by A X A 2 • • • A m . If a ls a 2 , •••, a m be any representatives of the classes A lt A 2 , • ••, A m , then a^-dm is a representative of the class A X A^ -'A m . If each of the m factors is the class, A, then the product is called the wth power of A and is denoted by A m . We have A 1 = A and A° = i. Theorem i. For every ideal class A there exists one and only one ideal class B such that the product AB is the principal class. 1 Let a be any ideal of the class A and a any number of a. Then ct6=(a), 1) where b is an ideal whose class we denote by B. Then from i) it follows that AB = i. 2) If now a class C other than B exist such that AC=i, 3) 1 Hilbert : Bericht, Satz 45. 28 434 THE IDEAL CLASSES OF A QUADRATIC REALM. we have from 2) ABC = C, and hence, making use of 3) B = C. The theorem is therefore proved. The class B is called the reciprocal class of the class A and is denoted by A- 1 . It is evident that inversely A is the reciprocal class of A' 1 . Defining further A- m as the reciprocal class of A m , the follow- ing laws are seen to hold for any positive integral rational expo- nents, r, s. A r A 8 = A r+s , (A r ) s = A rs , (AB) r = A r B r . Theorem 2. // A be any ideal class and b any ideal, there exists in A an ideal prime to b. 1 The quotients obtained by dividing each number, a, of an ideal a by a are evidently ideals that belong to a single class. Among them can be found an ideal prime to any given ideal b, for a can be chosen so that (a) /a is prime to b. Hence the theorem. § 3. The Class Number of a Quadratic Realm. We shall now show that the number of ideal classes in any given quadratic 2 realm is finite; that is, there exists in every quadratic realm a system of ideals finite in number such that the product of any ideal of the realm by one and only one of these ideals is a principal ideal. Such a system of ideals for a given realm we shall call a complete system of non-equivalent ideals. The number of ideals composing such a system, that is, the number of ideal classes of the realm is denoted by h. To prove that h is finite we need the following theorem: Theorem 3. In every ideal a there exists a number a different from and such that 1 Dirichlet-Dedekind : p. 579. 2 This theorem holds for the general realm of the wth degree. THE IDEAL CLASSES OF A QUADRATIC REALM. 435 |*MUI»MV<*|, zvhere d is the discriminant of the realm. 1 We shall distinguish two cases according as the realm is real or imaginary. i. Let a be any ideal of a real quadratic realm, k, and a basis of a, where o) 1 , w 2 is a basis of k. Since a lt a 2 and their conjugates, a/, a/ are real numbers, k being a real realm, Ojjr + 3 2 3^ a /' r + a 2 r 3' are linear forms with real coefficients, and their determinant can easily be shown to be different from o. Hence by Minkowski's Theorem (Chap. XIII, Th. i) there exist rational integers, x , y , such that K'.r + a 2 'y \ g IV^a/ — a 2 a/| It is easily seen that a, = a 1 x -\-a. 2 y , is the desired number of a, for if a — a x x -\-a 2 y , then a' = a 1 'x -\- a 2 y , and hence from i) Moreover, that is, «i a 2 = a x a 2 *>1 *>2 and hence \afi % t OA'I- = / [a]V*|, |»[a]|s|»[o]V?|. ii. The realm is imaginary. Let a x = pi + foj, a 2 = p 2 + *<t 2 , where p a , p 2 , o-j, <r 2 are real numbers and i = V — i, be a basis of a. Since pi, p 2 , o^, o\> are rea l numbers, whose determinant is dif- ferent from o, there exist by Minkowski's Theorem rational in- tegers, x , y , such that |Pl- r + P23'o| ^ I VPl^2 /^l oV^o + <r 2 J 1 g j Vpi°2 — P 2 ^i 1 Hilbert : Bericht, Satz 46. 436 THE IDEAL CLASSES OF A QUADRATIC REALM. We shall show that a = a x x + a 2 y is the desired number. We have a =ct 1 x +a 2 y ^p^o + PsS^'fo^o + »*?•)» a' = ck'x 9 + a 2 'y = Pl x + p 2 y — $ ( <r t *t + * 2 y ) , n[a] = ( Pl x + p 2 y ) 2 + (a t x 9 + <r 2 y Y, and hence M M ^2| Pl o- 2 — p 2 (T x \. It is easily seen, moreover, that I «ia 2 ' — a 2 a/ 1 = 2 1 Pl (r 2 — p^ | , whence «[a] ^ [flA' — a 2 a x '|. We have, however, as in i, \a x a 2 ' — a 2 a 1 '\ = \n[a]Vd\, and therefore n[a] g |»[q] y~d\. Theorem 4. There exists in every ideal class of a realm, k, an ideal whose norm does not exceed the absolute value of the discriminant of k} Let A be any ideal class and \ an ideal of the reciprocal class A- 1 . By the last theorem there exists in j a number, 1, such that |«H1*|»fflV3|. 2) But (0=Kt, 3) where a is an ideal belonging to the class reciprocal to A' 1 ; that is, to A. From 3) it follows that \n[ L ]\=n[i]n[a], and hence from 2) »MslV3|- 1 Hilbert : Bericht, Satz 50. THE IDEAL CLASSES OF A QUADRATIC REALM. 437 Theorem 5. The number of ideal classes of any realm is finite. 1 Since every ideal is a divisor of its norm, we shall by the last theorem obtain at least one representative of each ideal class of any given realm, k ; that is, a complete system of non-equivalent ideals, if we resolve into their ideal factors all positive rational integers which are less than |V^|> where d is the discrimi- nant of k. There are evidently only a finite number of rational integers satisfying this condition and each of them is resolvable into only a finite number of ideal factors. The number of ideals of k whose norms are less than |V^| * s therefore finite. Hence the number of ideal classes of k is finite. The last two theorems enable us to determine the number of ideal classes of any quadratic realm, the method consisting sim- ply in determining into how many classes the finite number of ideals fall, whose norms are less than |V^|. 2 We shall illustrate this method of determining the class number by several examples. This we do the more readily as in the solutions of these examples will be found many of the problems which arise in reckoning with ideals. Our task then being to ascertain into how many classes the ideals of any given realm, k, fall, whose norms are ^|V^|, it is evident that this will be accomplished, if we determine into how many classes fall the prime ideals and those of their powers and products whose norms satisfy the given condition. Having determined the prime ideals whose norms are g|V^| by resolving all rational primes which are g|V^| m *o their ideal factors, we next determine what equivalences exist between these ideals, including, of course, (1) as a representative of the prin- cipal class. The number of classes given by these prime ideals and (1) having been determined, it remains to be ascertained x Hilbert: Bericht, Satz 50. 2 This method of determining the class number of a realm is applicable to realms of higher degree. See Hilbert : Bericht, p. 226; also " Tafel der Klassenanzahlen fur Kubische Zahlkorper " by the author. 438 THE IDEAL CLASSES OF A QUADRATIC REALM. whether any powers and products of these prime ideals, the norms of such powers and products being ^| \/d\, give new classes. The solution of the question whether or no two given ideals are equivalent will be discussed in full in connection with the numerical examples. Theorem 6. // h be the class number of a realm, k, the hth power of every ideal class is the principal class. 1 Let A be any ideal class of k. In the series A, A 2 , • ••, A r , • ••, we must have two classes the same, as A r+e = A r , and hence A e = I. If A e be the lowest power of A which gives the principal class; then the classes A°=i, A, A*, .-., A+* 4) are all different. If B be a class different from all the classes 4), then the classes B, AB, A 2 B, -.., A^B are all different from each other and from each of the classes 4). Continuing this process, we see that h is a multiple of e. But e was the exponent of the lowest power of any class that gives the principal class. Hence the hth power of every class of k is the principal class. From this theorem it is evident that the hth power of every ideal is a principal ideal. Ex. 1. k(i). Basis: 1, i. d = — 4. Each class must contain an ideal whose norm is ^ | \/ — 4 |, that is ^2. We shall indicate this by writing n[ct] i | V — 4l> w[ct] =1 or 2. We have (2) = (l + 2 . The only ideals whose norms satisfy the given condition are therefore (1) and (i-f-*)» both of which are principal ideals. There is therefore 'Hilbert: Bericht, Satz 51. THE IDEAL CLASSES OF A QUADRATIC REALM. 439 only one class, the principal class. Hence h = 1. Therefore the ordi- nary unique factorization law holds in k(i), as we have already seen to be the case. Ex.2. k(\/~^z). Basis: 1, i(i + V" 1 ^)- d = — 3 n[a]^\ V^3|, »[<*] = 1. The only ideal whose norm satisfies the given condition is 1, hence there is only one class, the principal class; that is, ftssrl. Ex. 3. *(N/2). Basis: 1, V 2 - d = 8 r e have n[a] ^ | V8 j, n[a] = 1 or 2, (2) = V2) 2 . The only ideals whose norms satisfy the given condition are (1) and (\/2), both of which are principal ideals. Hence ft ±3 1-. Ex. 4. &(V — 5)- Basis: 1, V— 5- d — — 20. w[a] g I V —20 |, n[a] = i, 2, 3, or 4. We have (2) = (2, i + V^5) 2 > (3) = (3, i + V^"5)(3, i-V^"5). We have now to determine what equivalences, if any, exist between the ideals (1), (2, I + V~ 5), (3, i + V^). (3, 1 — V^5) and (2, 1 + V — 5) 2 > these being all the ideals whose norms satisfy the given condition. We see at once that (2, 1 + V — 5) 2 , = (2), is a principal ideal and represents therefore with (1), the principal class. On the other hand, it is easily shown that (2, 1 -J- V — 5) is a non- principal ideal, for, if it were a principal ideal, there must exist an integer, a, = x + yV — 5, such that (a) = (2, i + V=S). and hence w[a]=w(2, 1 + v^s); that is, two rational integers, x, y, must exist such that This is, however, manifestly impossible. Hence (2, 1 + V — 5) * s a non-principal ideal and the representative of a new class, which we shall denote by A. We have already proved (§ 1) that (3, 1 + V — 5) ar, d (3, 1 — V — 5) are equivalent to (2, 1 + V — 5). They belong therefore to A, and all ideals of &(V — 5) fall into two •classes, 1 and A. Hence h = 2. It will be observed that A*=l. 440 THE IDEAL CLASSES OF A QUADRATIC REALM. Ex. 5- *(V7>, Basis: 1, \f?. d = 28. «[<*] ^ I V28 I, n[a] = 1, 2, 3, 4, or 5. We have (2) = (2,1+ V7) 2 (3) = (3, i + V7)(3, i-V7) (5) = (5)- 1 The ideals to be considered are therefore (1), (2, i-j-\/7)> (3» I + V7)» (3, 1- V7), (5) and (2, 1 + V7) 2 ; of these (1), (5) and (2, 1 + V7) 2 belong to the class 1. We proceed as in the case of (2, 1 -f- V — 5) in the last example to determine whether (2, 1 -\- V7) i s or * s not a principal ideal. In order that (2, 1 + V7) ma y De a principal ideal, it is necessary and sufficient that there exist an integer a, = x -f- y yjj, such that \n[a] \=n(2, 1 + V 7 ) ; that is, that there exist rational integers x, y, such that x 2 — 7y 2 = 2 or — 2. We see that x = 3, y = 1 satisfy this condition. 2 Hence (2, i + V7) = (3 + V7), a principal ideal, 3 + V7 being divisible by (2, 1 + V7), since the latter is the only ideal whose norm is 2. We can in like manner show that (3> 1 + V7) i s a principal ideal, for x = 2, y = 1 satisfy the condition * 2 — 7y 2 = — 3 whence (3, 1 + V~7) = (2 + V7) or r 2 — V7). So far as the task in hand is concerned, it is indifferent to which of the two conjugate principal ideals, (2 + V7) an d (2 — V7)» (3> 1 + V7) * s equal, for all that we need know is that it is a principal ideal, from which it follows at once that (3, 1 — \/7) 1S a principal ideal, for it belongs to the class reciprocal to that of (3, 1 + V7) since (3, 1 + V7) (3, 1 — V7) — (1). It is easily seen, however, that 2 + V7 is not a number of (3, 1 + V7) while 2 — \/7 does enjoy this property. Hence (3, i + V7) = (2-V7), and (3, i-V7) = ( 2 + V7). All the ideals of k{\Jj) whose norms are i | V^ I being principal ideals, we have h = 1. 1 This denotes that (5) is a prime ideal. 2 See also Chap XIII, § 5, Pell's Equation. THE IDEAL CLASSES OF A QUADRATIC REALM. 44 1 We are assisted in determining to which of the classes, i, A, A 2 ,.-., A 1 , if any, a given ideal j belongs by the following theorem : Theorem 7. // a' be the lowest power of a which is a prin- cipal ideal, a, a 2 , • • •, a' *-> 1, being representatives of the t classes A,A\ ■•',A*=i, 5) and } 8 the lowest power of an ideal j which is a principal ideal, then in order that j may belong to one of the classes 5) it is neces- sary that t shall be divisible by s, and furthermore, if this condi- tion be satisfied and t=t 1 s, then \ can belong to none of the classes 5) except the <f>(s) classes A*, for which i=i 1 t ly and i x is prime to s. If i-a% then j* ^ a** *** I ** j 8 , whence £ = 0, mod s; that is, t divisible by s is a necessary condition that j shall belong to one of the classes 5). Furthermore, if * j r-* a\ then I s ~> a 8i r^x^a* t whence si sb 0, mod t, sss t^, and therefore i as 0, mod 1 1 ; that is, i = i x t x . Then j ^ qMi, f^a 2i ^, jf , — ■ a fiiti , \ 9 — ' a ffhti , 442 THE IDEAL CLASSES OF A QUADRATIC REALM. from which it follows, since no two of the ideals j, j 2 , • • • , j s are equivalent, that must be incongruent each to each, mod t ; that is, we must have where / and g are any two of the integers, i, 2, •••, s, different from each other. Therefore we must have fi x =^gi x , mod s; that is, the integers i lt 2$ lt •••, si t must form a complete residue system, mod s, which can be the case only when i 1 is prime to ,?. Hence in case j should belong to any one of the classes 5) it is possible only to have i ~ a*** 1 , where t x = t/s, and i 1 is prime to s. There are therefore only <f>(s) of the classes 1) to which it is possible for j to belong. Ex. 6. Let a 2 * be the lowest power of a which is a principal ideal, a, a 2 , •••, cf*. — 'I, representing therefore the twenty-four classes A, A 2 , -.., A 2i =i, 6) Let f be the lowest power of j which is a principal ideal. Since 24 is divisible by 6, it is possible for j to belong to 0(6) =2, of the classes 6). We have £ = 4, and those of the classes 6) to which it is possible for j to belong are A* and A 20 . By means of Th. 7 we can reduce the labor of determining h ; for, if a be an ideal satisfying Minkowski's condition, that is, n[a] ^|V^|j an d Q* the lowest power of a that is a principal ideal, then a, a 2 , •••, a*^ 1, are representatives of £ ideal classes, ^, ^ 2 , ...,^*=i, 7) and, as we have seen in the last theorem, h is a multiple of t. Let now N be the number of ideals of the realm that satisfy THE IDEAL CLASSES OF A QUADRATIC REALM. 443 Minkowski's condition, n the number of these ideals that belong to one or the other of the classes 7), and c the number of the known classes 7) that have found representatives among the ideals satisfying Minkowski's condition. The t classes 7) must evidently have representatives among the A r ideals satisfying Minkowski's condition, and therefore, since only c of these classes have yet found representatives among these ideals, t — c of the JV — n of these ideals whose classes have not yet been determined must belong respectively to the t — c classes whose representatives are missing. We have then as possible representatives of new classes N — n — (t — c) ideals, and, if N — n — (t — c)<t; that is, if N — n-\-c<2t, it follows, since h must be divisible by t, that h = t. In particular, if N < 2t, we have at once h = t. If A r — n + c<£2t, we must proceed to determine whether some of the remaining ideals belong to the classes 7). Let j be one which is found to belong to none of the classes 7) and let \ 8 be the lowest power of j which is a principal ideal. Then j, j 2 , •••, j* _1 are representatives of the s — 1 new classes, B, B 2 , • • • , 5 S_1 , and there are now in all st known classes 1, A, A\ ..-, A** 9 B, BA, BA 2 , •••, BA*-\ 8) B*-\ B*~K4, B^A 2 , ••-, £ s -M'-\ and h is therefore divisible by st. 444 THE IDEAL CLASSES OF A QUADRATIC REALM. If now n and c have their former meaning except that 8) are now the known classes, and if N — n + c < 2st, then h = st. If, however, N — n + c <£ 2^, we proceed as before to determine the classes to which the remain- ing ideals belong, observing always whether N — n + c < 2tf. If we find one that belongs to none of the classes 8), we proceed as with j. fl/-3i Ex.7. KV—li). Basis: i,—^- ±,d=- 3 »[<*] = IV — 31 I; »[<*] = 1, 2, 3, 4 or 5. We have (3) = (3), (s) = ( s . Lbfc*) (s.5=^> Since ^ + ^ + 8/ =4=2 for any integral values of x and y, there is no integer of &(V — 3 1 ) whose norm is 2. Hence ( 2 ,i+^.)~,( We proceed to determine the lowest power of ( 2, j that is a principal ideal. We have since the only integer of &(V — 3 1 ). whose norm is 4, is 2, and, if then (,i±^ii)= ( ,), which is impossible. THE IDEAL CLASSES OF A QUADRATIC REALM. 445 We have since 8 3= l + V— 3 1 . l — V — 3\ 2 2 Hence we have so far found representatives, 1, ( 2, 5l J , a nd ^i+J^ZilV ( 2y 1 + ^~ 3I V^iof three classes 1, A, A 2 , {A 3 =i). Therefore h is divisible by 3. Of the eight ideals satisfying Minkowski's condition, (i),( 2, — ■ — J, and ( 5> J four belong to these classes and from we see that ( 2, ^ J belongs to A 2 , and hence ( 2, £ 5- J to A. The inequality N — n + c < 2t is now seen to hold, for we have N = 8, n = 6, c = 3, and * = 3, and it is evident that h = 3. The classes to which ( S> ^ ) and ( 5, — ] belong are easily determined, since and 3-f V ~ Zl is a number of both (2, 1 ~ V ~ 3l \ and (5,-^—). whence Therefore (5> 3 + ^~ 31 ) belongs to A, and ^IZUtlHj to A 2 . Ex. 8. KV82). Basis: 1, V82. ^ = 328. «[«] = V328 I ; n[a] = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, II, 12, 13, 14, 15, 16, 17, or 18. 446 THE IDEAL CLASSES OF A QUADRATIC REALM. We have (2) = (2, V82) (2, V&) (3) = (3, 1 + V82X3, 1-V82) (5) = (5) (7) = (7) (Il) = (ll, 4+ V82)(n, 4-V82) (13) =(13, 2 + V82)(i3, 2 — V&) (I7) = d7). We must now determine whether (2, V82) is a principal ideal. To do this we determine whether £(\/82) contains an integer whose norm is 2; that is whether integral values of x and y can be found satisfying the equation x 2 — S2y 2 = 2. 9) Using Th. 6, Chap XIII, and developing V82 as a continued fraction, we see that 1 1 y / 82 = 9 -f 18+18-^ and have n a n Pn fn M n " 9 9 I I 2 18 163 18 I From this it is evident that 9) has no solution, and hence that (2, V82) is a non-principal ideal. From this development of V82, it is also evident that k(yJ82) contains no integers with norms 3, 5, 6, or 7, and furthermore 9 + V82 is the fundamental unit. That &(V82) contained no integers with norms 5 or 7 was, of course, already shown by the fact (5) and (7) are principal ideals. We have, however, learned, in addition to the fact that (2, V82) is a non-principal ideal, that (3, 1 -f- V82) and (3, 1 — V82) are non-principal ideals, since &(V82) contains no integer with norm 3, and, moreover, that neither of the products of these last two ideals by (2, V82) can be a principal ideal, since &(V82) contains no integer with norm 6. We shall now determine into how many classes the ideals, which have been proved to be non-principal, fall. We have (2, \/82) as a representative of a new class, A, and A 2 = 1. Calculate now the norms of a few integers of fc(V82). We have w[8+ V82]= — 18. Hence (18) is the product of three ideals whose norms are 2, 3 and 3 respectively. Since 8+V82 is a number of (3, 1 — V82) and not of (3, 1 + V82), we must have (18) = (2 f V82) (3, 1-V82) 2 . THE IDEAL CLASSES OF A QUADRATIC REALM. 447 From which it follows that (3, I — V&O 2 belongs to A, and (3, 1 — V82) gives a new class B. We have A = B 2 . But n[i — V82] = — 81= — 3*, and 1 — V& 2 is a number of (3, 1 — V& 2 ) and not of (3, 1 + V& 2 ). Hence (i-V82) = (3, 1-V82)*, and we see that we now have four classes 1, B, Br, B 3 (£* = 1), as repre- sentatives of which among the ideals satisfying Minkowski's condition, we may take (1), (3, 1 — V82), (2, V&O and (3, 1 -f V82). We have now A' = 28, n = 24, c = 4, and f = 4, and hence AT — n -f c <£ 2t ; that is, there are four ideals, the factors of (11) and (13), whose classes are yet undetermined and we have found representatives of all of our four known classes. One of these remaining ideals might therefore give a new class and we should have h = 8. That h is either 4 or 8, we now know. This is, however, easily settled, for n{y -J- V82I — — ^ and 7 + V82 is a number of both (3, 1 + V82) and (11, 4 — V82). Hence (7 + V82") = ( 3 , 1 + V82K11, 4-V&), and (11, 4 — V82) belongs to the class B. Therefore h = 4 - We see that (11, 44-V82) belongs to B 3 and from the fact that n[2+V82] = — 78 = — 2 • 3 • 13, we can show easily that (13, 2 -f V82) belongs to B and (13, 2 — V82) to B 3 . Ex. 9. Show that h=z 6 for k (V — 26), h = 1 for &(V — 19), /i = 2 for ^(V I 5)> h = 2 for &(\/26), A = 4 for &(V — 34), h = 6 for fc(V^6i). The labor of finding h by this method can be reduced by using another theorem, due also to Minkowski, which gives a smaller limit below which the norms of the representatives of the classes must fall, thus diminishing the number of ideals to be examined. This theorem for the general realm of the nth degree is as follows : In every ideal class there is an ideal, a, such that ■M<(;?)'5!l»?l' where n is the degree of the realm, d its discriminant, and r the number of pairs of imaginary realms which occur among the conjugate realms, In a real quadratic realm, we have n[a] < I \ yjd ' \, and in the case of fc(\/82) need, therefore, to examine only those ideals whose norms are less than 10. It will be noticed that we did find, as representatives of all classes, ideals whose norms satisfied this condition. 1 Minkowski : Diophantische Approximationen, p. 185. See also " Tafel der Klassenanzahlen fur Kubische Zahlkorper " by the author for its application to cubic realms. 448 THE IDEAL CLASSES OF A QUADRATIC REALM. For a table giving the class numbers of quadratic realms, their funda- mental units and other data, see J. Sommer: Vorlesungen uber Zahlen- theorie. This table extends, for imaginary realms, to ra = — 97, and, for real realms, to m = 101. This book should be consulted by those who wish to pursue the subject further. The class number of a realm can also be expressed by means of an infinite series. See Hilbert : Bericht, Cap. VII and §79; also Dirichlet- Dedekind: §184. We shall close this chapter with a theorem that gives important information regarding the class number of a realm in a certain special case. For its proof, we shall need two theorems, the second of which throws additional light upon the question whether the norm of the fundamental unit of a real quadratic realm is 1 or — 1. Theorem 8. Every number, a, of a quadratic realm, &(\/m), whose norm is 1, can be represented as the quotient, y/y', of two conjugate integers, y, y , of the realm} We have seen that a can be put in the form a + bat c where 1, w is a basis of the realm and a, b and c are rational integers. Let y — x-\- ym, where x and y are rational integers to be determined, and let the rational equation of which w is a root be x 2 + ex + f = o. Put a -f- boy x 4- \'o) N ! —g — L^_ . 10) C X + To/ Making use of the relations »-)-«'== — e, and ww' = /, we have from 10), as the equations that x and 3! must satisfy, (o — c)x-\- (bf — ae)x = o, ") bx — (a + c)y=0. These equations evidently have a solution different from x = (\ 1 See Hilbert : Bericht, Satz 90. THE IDEAL CLASSES OF A QUADRATIC REALM. 449 y = o when and only when the determinant, D, of their coefficients is o, and, if D = o, they have an infinite number of solutions jfss=:r.Xii y = ryi, where x x , y t is any particular solution different from o, o, and hence have an infinite number of integral solutions, for we can choose r so that rx lf ry ± are integers. We have D = — a 2 + abe — b 2 f + c 2 = — n [a] • c 2 + c 2 = o, since n[a] = I. Hence the equations n) have an infinite number of integral solutions and the theorem is therefore proved. - As a particular solution of n), we may take x = a-\-c, y = b, all integral solutions then being of the form s(a + sb *-._-■ , y= -j, where ^ and t are rational integers and t a common divisor of a-\- c and b. We can, of course, take a, b and c without a common divisor, and then have also a prime to b, since n[a] = i. Ex. Let a = — . We have a = 2, b=i, c = 3, and hence 3 3 5— V"— S Theorem 9. // f/i^ discriminant, d, of a real quadratic realm, k(y/m), be divisible by a single prime number, the norm of the fundamental -unit of the realm is — I. 1 In order that d may be divisible by a single prime number, we must have m = 2, or a prime = 1, mod 4. Let c be the fundamental unit of &( V w )- If w[e]=i, by Th. 8 there would exist an integer, y, of k(-\/m) such that e = ?-. 12) r Then from 12) it would follow that (y) = (/), 1 Hilbert : Bericht, p. 294. 29 450 THE IDEAL CLASSES OF A QUADRATIC REALM. and hence that (y) is either an ambiguous ideal (p. 347), an ambiguous ideal multiplied by a rational principal ideal (a), or (a). Since, however, d is divisible by the single prime m, the realm contains only one ambiguous prime ideal (\/w), which is therefore the only ambiguous ideal of the realm. Hence, we must have (y)='(Vw), (aV») or (a), and therefore y^^ym, rja^/tn or rja, where 77 is a unit. But we have then from 12) or and hence c = — rj 2 or rf, from which it would follow that e is not the fundamental unit, as was assumed. Hence the assumption that n [e] = 1 is untenable, and the theorem is proved. The realms k{^/2), k{-\J$) and &(\/i7), whose fundamental units have been found to be 1 + y 2, £(i + \/5) and 4+^17 respectively, will illustrate the truth of this theorem. Theorem 10. // the discriminant of a quadratic realm, k ( ^/m), be divisible by a single prime number, the class number, h, of the realm is odd. 1 Assume h to be even. Then there is in the realm certainly one non-principal ideal, j, whose square is a principal .ideal ; that is, j 2 r^ 1. But we have also jj' — 1, and hence j ^ f ; that is, there exist integers, a, /?, of the realm such that (a)i=(/J)f. 13) From 13) we have n[(a)]=n[(p)] 9 whence a//3, = /c, is a number of the realm whose norm is ± 1. When k(-\/m) is imaginary, we have *[*]=* I, and when k(\/m) is real and n[e] = — ■ 1, where e is the fundamental unit, we have either n[K.]=ii, or n[ac] = i. By Th. 8 we can put K = y/y, or CK = y/y', according as h[k] = 1 or. — 1, y and y' being conju- gate integers of the realm. In both cases, we have 'Hilbert: Bericht, Hiilfsatz 13. THE IDEAL CLASSES OF A QUADRATIC REALM. 45 I and hence from 13) (y)j= (y')i'» as a consequence of j 2 <-*> 1, where j is a non-principal ideal ; that is, as a consequence of h even. Hence (y)j is either an ambiguous ideal, an ambiguous ideal multiplied by a rational principal ideal (a), or (a). Since, how- ever, when m = 2, or a prime = 1, mod 4, the realm contains no ambiguous ideal other than (Vm) (see proof of Th. 9), and, in in the case of k(i), the only ambiguous ideal is (1 + *). We see that in all cases (y)j must be a principal ideal, and hence j a principal ideal. But this renders untenable our assumption that h is even. Hence h is odd. The realms k(i), k(^/ — 3), k(y f 2) and k{^J — 31), whose class numbers were found to be 1, 1, 1 and 3 respectively, will illustrate the truth of this theorem. It is evident that in determining the class number of a realm, satisfying the conditions of Th. 10, we can use, since h must be odd, instead of the inequality N — n -j- c < 2t, the inequality N — n -f- c < 3/, thus shortening the work still further. Making use of this in Ex. 7, it is unnecessary to determine the class to which belongs ( 2, j. INDEX. Numerals refe Ambiguous ideal, 347. Appertains, exponent to which an in- teger, 99, 393. Associated integers, in R, 9 ; in k(i), 163; in k(V — 3), 223; in kiV^), 246. Basis, of k(i), 159-161 ; of £ V — 3), 220; of k(V2), 232; of k(V — 5), 245 ; of k(Vtn), 284-287, determi- nation, 289-292 ; of ideal, 293-295, determination, 351-355. Biquadratic residues and reciprocity law, 205-217. Character of an integer, quadratic, in R, 121, in k(i), 212; biquadratic, 209, 212 Classes, ideal, definition, 432 ; prin- cipal class, 432 ; product of, 432 ; reciprocal, 434. Classification of the numbers of an ideal with respect to another ideal, 326-330. Class number of a realm, definition, 434; is finite, 437; determination, 437-448, 45 1- Congruences, definition, 31, 297, 323 ; elementary theorems, 32-37 ; 323- 326 ; of two polynomials, 57, 370 ; of condition, 59-61, 369-372 ; of first degree in one unknown, 68-70, 375-38o ; equivalent, 62-64, 372, 373 ; transformations, 62-64, 372, 374 ; equivalent systems, 64 ; of nth degree in one unknown, preliminary discussion, 66-68, 374, 375, root, 66, 374, with prime modulus, 88-90, 385-387, composite modulus, 95-97, 39i, 392; multiple roots, definition, 452 r to pages. 89, 386, determination, 93, 94, 386 ; limit to number of roots, 89, 386 ; x4><-m> — i = o, mod m, 90 ; x<t>( m ) — 1 == o, mod m, 387, 388 ; common roots, 92, 93, 389; binomial, 110- 112, primitive and imprimitive roots, in; x n ^=b, mod p, 114- 116, Euler's criterion, 115; of sec- ond degree with one unknown, 119- 121 ; solution of x 2 ^^. — 1, mod p, by means of Wilson's theorem, 129, 130; in k(i), 180, of condition, 190. Conjugate, numbers, 4 ; realm, 4. Dirichlet's theorem regarding infinity of primes in an arithmetical pro- gression, 11. Discriminant, of k(i), 161 ; of k(V — 3), 221; of k(y/2), 232; of k(V^S), 245; of k(Vm), 287, 288 ; of number, 284. Divisor, greatest common, in R, 16, 18, 25; in k(i), 173; of two ideals, 310-313, 318; discussion of defini- tion, 252. Divisors, of integers in R, number of, 23, sum of, 24 ; of ideal, number of, 318. Equivalence of ideals, 427-431 ; in narrower sense, 431. Eratosthenes, sieve of, 10. Euler's criterion for solvability of xn^=b, mod p, 115, 122. Factorization of a rational prime de- termined by (d/p), in k(i), 179; in k(V — 3), 229; in fe(Vm), 347, 348. Fermat's theorem, 57 ; as generalized INDEX. 453 by Euler, 57 ; analogue for k(i), 189 ; analogue for ideals, 368, 369. Frequency of the rational primes, II, Galois realm, 281. Gauss' lemma, 130. Generation of realm, 3. Ideal numbers, necessity for, 253 ; nature explained, 254-257 ; Kum- mer's, 267. Ideals, definition, 257, 293 ; numbers of, 293 ; basis of, 293-295 ; can- onical basis of, 294 ; determination of basis, 298-301 ; numbers defin- ing, 295 ; symbol of, 257, 295 ; in- troduction of numbers into and omission from symbol, 258, 295, 296 ; principal and non-principal, 260, 261, 297 ; conjugate, 301 ; basis of conjugate, 301 ; equality of, 258, 259, 302 ; multiplication of, 261, 262, 302, 303; divisibility of, 263, 303 ; common divisor of, 303 ; prime, 263-265, 304 ; norm of, 326- 338, 351. Imprimitive numbers, see primitive numbers. Incongruent numbers, complete sys- tem of, in R, 34; in fe(i'), 182-185; in k(Vm), 326. Index, of a product, 106, 399; of a power, 106, 399. Indices, definition, 105, 399; system of, 106, 399 ; solution of congru- ences by means of, 1 08-1 10, 400- 402. Integers, of R, 7, 23; absolute value in R, 7 , 33; of k(i), 157; of k(V —3), 219; of k(V2), 231; of k(V — 5), 245; of k(Vm), 284- 287; general algebraic, 1, 275-279. Legendre's symbol, 127. Multiple, least common, in R, 25 ; of two ideals, 310-312, 318. Non-equivalent ideals, complete sys- tem of, 434. Norm, of a number, in fe(i'), 156; in k(V^3), 218, 221 5 in k( V2), 231 ; in k(V — 5), 245; in k(Vrn), 283; of an ideal, definition, 326, 337, value, 330, determination, 351 ; of a product of ideals, 334 ; of a prin- cipal ideal, 337 ; of a prime ideal, 338. Numbers, algebraic, definition, 1 ; de- gree of, 1 ; conjugate, 4 ; rational equation of lowest degree satisfied by, 2, 273; of R, 7; of k(i), 155; of k(V~^3), 218; of fc(V2), 231; of k(V — 5), 245; of the general realm, 271-279; of fe(Vw), 281. Number class, rational modulus, 32, 33 ; ideal modulus, 324. Pell's equation, 423-426. 0- function, in R, definition, 37, gen- eral expression, 38, 44, 53, product theorem, 45, summation theorem, 46, 75, of higher order, 54 ; in k(i), 185-188; for ideals, definition, 358, expression for power of prime ideal, 359, general expression, 359-- 362, 366, 367, summation theorem, 362, 363, 367, product theorem, 360, 361, of higher order, 367. Polynomials in a single variable, 268- 271. Polynomials with respect to a prime modulus, reduced, 62 ; degree of, 76 ; divisibility of, 76, 380 ; com- mon divisor of, 76, 380 ; common multiple of, 76, 380 ; unit, 77, 381 ; associated, 77, 381 ; primary, 78, 381 ; prime, 78, 381 ; determina- tion of prime, 78, 381, 382; congru- ence with respect to a double modulus, 81 ; unique factorization theorem for, 82-87, 382-385 ; divi- sion of one by another, 382. Power of a prime by which m ! is divisible, 26. 454 INDEX. Primary integers of k(i), 193-196. Prime factors, resolution of an ideal into, 348-350. Prime ideals, of k(V — 5), 263-265; of k(Vm), definition, 304, deter- mination and classification, 339- 348. Prime numbers, of R, definition, 9, infinite in number, 10 ; of k(i), defi- nition, 165, classification, 177; of k(V — 3), definition, 223, classifi- cation, 227-230; of k(V2), defi- nition, 235, classification, 238-240 ; of k(V — 5), 246, 247. Primitive numbers, of k(i), 157 ; of k(V~^3), 218; of the general realm, 274, 275; of k(V*n), 282, 283 ; with respect to a prime ideal modulus, 398. Primitive root, definition, 100; deter- mination, 112; of prime of form 2 2 4-i, 151; of prime of form 49+i is 2, 152. Realm, definition, 3 ; generation, 3 ; degree, 4 ; conjugate, 4 ; number defining, 4, 280 ; number generating, 4. Reciprocity law, for quadratic resi- dues, in R, 135; in k(i), 201-205; determination of value of (a/p) by means of, 144; other applications of, 149; for biquadratic residues, 210, 215-217. Residue, odd prime moduli of which an integer is a quadratic, 128, 145, 147 ; prime moduli of which — 1, is a quadratic, 128; prime moduli of which 2 is a quadratic, 133. Residue system, complete, in R, 33, 34; in k(i), 182-185; in fe(Vm), 326; reduced, in R, 37; in k(i), 185, in fe(Vw), 358. Residues of powers, definition, 98, 392 ; complete system of, 98, 393 ; law of periodicity, 100. Residues, n-ic, 116; quadratic, in R, 131, in k(i), 196-201 ; quadratic non-, 121 ; determination of quad- ratic, 124 ; with respect to a series of moduli, integer having certain, 70 ; cubic, 250 ; biquadratic, 205- 217. Sub-realm, 157. Symbol, Legendre's, 127 ; for ideal, 257, 295. Unit ideal, of k(V — 5), 263; of k(Vm), 304. Unit, fundamental, of fe(V2), 233; of k(Vm), definition, 420; deter- mination, 420-426. Units, of R, 8; of k(i), 163; of fc(V — 3), 222; of fc(V2), 232- 23S; of k(V — 5), 246; of fe(Vm), definition, 403, realm imaginary, 404, realm real, 405-426. Unique factorization theorem, in R, 12; in k(i), 167, 174, graphical discussion of, 169; in k(V2), 236, 237; in k(V—3), 226; in k(V — 5), failure of, 247-253, necessity for, 253, restoration in terms of ideal factors, 265, 266 ; realms in which original method of proof holds, 248-250; for ideals in fe(Vw), 305-317. Wilson's theorem, 91 ; as generalized by Gauss, 91 ; analogue for ideals, 388, 389- OF THE UNIVERSITY OF KTIUKIN TO DESK *KOM WHICH BUKKUWUD This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 5Jan'36JPY \30ec' 60 IF ! RZC'D LD OCC 2 2 19S5 C0 zap'iwi REC'D LD MAY 2 8 1957 *& R;^..„ MAY 1 9 195 9 ^- C REC'D LP MAY 3 1 1960 19Jan'62jH REC'D L D JAN 1 6 1962 DAVIS NTERUBRARY LOaJt < * **■* SIP* i 1974 l^lERUft NOV 7 74 PAVIS KARY LOAM JAN 3 1 a/3 ^•Ott-ftBU- ? n LD 21-100m-2,'55 (B139s22)476 General Library University of California Berkeley 329 * 20 e 574 UNIVERSITY OF CALIFORNIA UBRARY